# Full text of "Lessons Introductory to the Modern Higher Algebra"

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About Google Book Search Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web at |http: //books .google .com/I 600047028R r t ., / r / • • / ^-' •' ^ e^ . • ;>■ S 600047028R e .?/ ^ ■ ■ I'-' -I . £ , ' ; s r !•■■■ LESSONS INTRODUCTORY TO THE MODERN HIGHER ALGEBRA. CAMBRIDGE : PRINTED BY W. METCALFE AND BON, TRINITY STREET AND ROSE CRESCENT. LESSONS INTRODUCTOKY TO THE MODERN HIGHER ALGEBRA. BY GEORGE SALMON, D.D., REQIUS PROFESSOR OF DIVINITY IN THE UNIVERSITY OF DUBLIN. FOURTH EDITION. HODGES, FIGGIS, AND CO., GRAPTON STBEET, SOOKSEllESa TO TBE UmVEBSITT, 1885, H • • * H ( *V..i •gjdnoava AADCLIFFE. TO A. OATLET, ESQ.. AND J. J. SYLVESTEB, ESQ., I BSG TO INSCRIBE THIS ATTEMPT TO RENDER SOME OF THEIR DIBOOVBRIES BETTER KNOWN, |n ^tknofokbgmjmt OP THE OBLIGATIONS I AM T7NDBR, NOT ONLT TO THEIR PUBLISHED WRITINGS, BUT ALSO TO THEIR INSTRUOTIVB CORRESPONDENCE. PREFACR The pressure of other engagements having prevented me from taking any part in the pre- paration of this new edition of my Higher Algebra^ I have to express my obligations to the good offices of my friend Mr. Cathcart in revising the work and superintending its progress through the press. A comparison of the number of pages will show that he has made several additions to the Contents of the last edition. These will chiefly be found in "Applications to binary quantics," which are now divided into Lessons xvii., xviii. GEO. SALMON. Trinity College, Dublin, May, 1885. CONTENTS. LBSSONL OSTIRMIKAKTS.— PRKUMINABT XLLUBTBATIOHB AND DSffUIITlOHS. rA«B Bole of sigDB ....... ft SylToiter's umbral notation ..... 8 LESSON n. BBDVOnOH AND CALCULATION OV DSnRMnrART8. ICinois [called by Jaoobi partial determinanti] • • .10 Szamples of rednction ...... 13 Product of diffeienoes of n qnantiti— ezpreiMd as a determinant • 1^ Bednction of bordered Heanans . . • • 17 Gontinnanta .....•• IB LESSON m. MULTIPLICATION OV DBTXRMINANT8. The theorem stated as one of linear transformation . . , 21 Extension of the theorem . . • • . • 22 Examples of multiplication of determinants ... 28 Product of squares of differences of n quantities . . .28 Badins of sphere drcumscribing a tetrahedron . • 26 Belation connecting mutual distances of points on a drde or sphere • 26 Of fiye points in space . . • • 27 SyWester's proof that equation of secular inequalities has all roots real • 2S LESSON IV. MINOR AND RECIPROCAL DBTBBMINANTB. Belations connecting products of determinants ... 29 Solution of a sfstem of linear equations . . .29 Bedprocal systems ...... 80 Kinors of reciprocal system expressed in terms of those of the origmal . 80 Minors of a determinant which yanishes .... 82 Forms for expaoiding a determinant of the fourth otder . 32 LESSON y. BYMMBTRIOAL AND BKSW BYMMITRICAL DSTBRMINANTS. Differentials of a determinant with respect to its constituents . . 84 If a symmetric determinant Yanishes, the same bordered is a perfect square 87 Skew symmetric determinants of odd degree Yanish . . .87 Of eYen degree are perfect squares .... 88 Nature of the square root [see Jacobi, OreUey ii. 854 j xxiz. 286] 89 Orthogonal substitutions . . . . . .41 Number of terms in a symmetrical determinant ... 45 b CONTENTS. LESSON VL DISORIMINATINa SYMMETRICAL DflTBRMnTANTS. New proof that equation of secular inequalities has all its roots real Sylvester's expressions for Sturm's functions in terms of the roots Borchardt's proof ..... LESSON vn, SYMMETRIO FUNCTIONS. Newton's formula for sums of powers of roots Improvement of this process . • . Determinant expression for sums of powers Bules for weight and order of a symmetric function . Formula for sum of powers of differences of roots Differential equation of functions of differences Symmetric functions of homogeneous equations Differential equation when binomial coefficients are used Serret's notation PAOI ■ 48 49 64 67 67 67 68 60 61 62 64 66 LESSON vni. ORDER AND WEIGHT OF ELIMINANTS Eliminants defined .....* Elimination by symmetric functions .... Order and weight of resultant of two equations Symmetric functions of common values for a system of two equations Extension of principles to any number of equations . LESSON II. EXPRESSION OF ELIMINANTS AS DETERMINANTS. Elimination, by process for greatest common measure . • . Euler's method ...... CfOnditions that two equations should have two common factors Sylvester's dialytic method ..... Bezout's method ....... Gayley's statement of it . Jacobians defined . . . - . Jacobian and derived equations satisfied by common system which satisfies equations of same degree ..... Expression by determinants, in particular cases, of resultant of three equations Gayley's method of expressing resultants as quotients of determinants LESSON X! DETERMINATION OF COMMON ROOTS. Expression of roots common to a system of equations by the difiSerentials of the resultant ...... Equations connecting these differentials when the resultant vanishes Expressions by the minors of Bezout's matrix General expression for differentials of resultant with respect to any quantities entering into the equations ..... G^eneral oondiuons that a system may have two oommon roots • 66 67 69 72 76 76 77 78 79 81 83 84 84 86 87 91 91 93 96 97 CONTENTS. XI LESSON XL DISCRIMINANTS. Older and weight of diacrimmants Discriminant expressed in teims of the roots Discriminant of product of two or more f onctionB • Discriminant is of form a^<p + a|'^ Formation of discriminants by the differential equation Method of finding the equal roots when the discriminant Taniahes Extension to any number of variables Discriminant of a quadratic function 99 101 101 102 108 104 106 107 liESSON XU. LINEAR TRANSFORMATIONS. Invarianoe of discriminants Number of independent invariants .... Invariants of systems of quantics Govariants ....*• Every invariant of a covariant is an invariant of the original Invariants of emanants are oovariants . . . Gontravariants ..... Differential symbols are contragredient to variables , jT^ + y*} + <bc* absolutely unaltered by transformation liized concomitants ..... Evectants ..... Evectant of discriminant of a quantic whose discriminant vanishes 108 110 lis 114 114 116 117, 120 119 120 121 122 128 LESSON xm. FORMATION OF INVARIANTS AND GOVARIANTS. Method by symmetric functions .... Concomitants which vanish when two or more roots are equal Method of mutual differentiation of covariants and contravariants . Differential coefficients substituted for the variables in a contravariant give covariants ...... For binary quantics, covariants and contravariants not essentially distinct Invariants and covariants of second order in coefficients • Cubinvariant of a quartic ..... Every quantic of odd degree has an invariant of the 4^ order . . Cubicovariant of cubic . . . • Method of the differential equation .... Weight of an invariant of given order Binary quantics of odd degree cannot have invariants of odd order Coefficients of covariants determined by the differential equation Skew invariants ...... Investigation of number of independent invariants by the differential equation Source of product of two covariants is product of their sources Cayley's definition of covariants ..... Extension to any number ol variables . 124 125 126 127 127 129 129 129 130 180 130 130 131 131 132 184 135 186 zu CONTENTS. LESSON XIV. STMBOLIOAL RBPRBSENTATiOIT OF OONCOUITAinn. KeChod of fonnation by deriyatiye STmbols Order of deriratiye in co^cients and in the vaiiables Table of inyaziantp of the third order Hermite's law of redprocitj ..... Deriyatiye qrmlx^ for ternary qnantios . • Symbols for eyectantB • . . . . Kethod of Axonhold and Clebsoh PA«B 187 140 141 142 144 146 147 LESSON XY. CANONICAL FORMS. Generality of a form examined by its number of constants . . 150 Bednction of a quadratic function to a sum of squares . . 161 Principle that the number of negatiye squares is unaffected by real substitution 151 Beduction of cubic to its canonical form .... 152 Discriminant of a cubic and of its Hessian differ only in sign • 153 General reduction of quantic of odd degree .... 158 Methods of forming canonizant ..... 154, 155 Clondition that a quantic of order 2n be reducible to a sum of n, 2n^ powers • 156 Canonical forms for quantics of eyen order . . . . 157 Canonical forms for sextic and octayic . • . • . 159 For ternary and quaternary cubics .... 160 LESSON XVL 8TSTBMS OF QUANTICS. OomfainaBts defined, diff eventlal equation satisfied by them Number of double points in an inyolution Geometrical interpretation of Jacobian Factor common to two quantics is square factor in Jacobian Order of condition that u + kv may haye cubic factor Nature of discriminant of Jacobian Discriminant of disoriminant of u + iEw Proof that resultant is a combinant . • Discriminant with respect to Xj y, of a function of u,v Discriminant of discriminant of iH-kv for ternary quantics Tact-inyariant of two curyes Tact-inyariant of complex curyes Osculanta . . - . Coyarionts of a binary system connected with those of a temaiy 161 162 162 162 163 164 165 166 167 169 169 170 171 172 LESSON xvn. APPLICATIONS TO BINARY QUANTICS. Inyariants when said to be distinct Number of independent coyariants Cayley's method of forming a complete system Thbquadric .... Besultant of two quadrics .... 176 176 177 178 180 CONTENTS. ••• XIU PAOB STBTBM 07 THBBB OR MORB QUADRICS . - . . . 181 KTtenffl'on to qualities in general of theorems conoeming qnadrios • 182 The CX7BI0 . . . . . . . 183 Geometric meaning of coTariant cubic .... 184 Square of this cubic expressed in terms of the other ooTaiiantB • 186, 192 Solution of cubic . . • • ■ • • . 186 SYSTBM 07 CUBIC AND QUADRIO .... 187, 225 Geometrical illustrations ...••. 189 Thb quartic ....... 189 Catalecticants . . . » . , 190 Discriminant of a quartic ..... 190 Belation of coyariants of cubic derived from that of invariants of a quartic . 191 Seztic covariant geometrical meaning of . . • 198 Relations connecting quadratic factors of . . . , 194 Beduction of quartic to its canonical form . . , 194 Belation connecting covariants of quartic . . • . 19ft Symmetrical solution of quartic ..... 196 Criteria for real and imaginary roots ..... 197 The quartic can be brought to its canonical form by real snbstitntiona . 197 Conditions that a quartic should have two square factors . . 198 Cayley's proof that the system of invariants and covariants is complete . 199 Application of Bumside's method ..... 200 Covariants of system of quartic and its Hessian . . . 201 Hessian of Hessian of any quantic ..... 203 SYSTBM 07 QUADBIC AlO) QUARTIC .... 202, 267 System 07 TWO CUBIC8 ...... 204 Besultant of the system ..... 20ft Condition that t» + Xv may be a perfect cube . . . . 20ft Mode of dealing with equations which contain a superfluous variable • 207 Jaoobian and simplest linear covariants .... 209 Any two cubics may be regarded as differential coefficients of same quartic . 210 Invariants of invariants of «+ Xr are combinants . . .211 Process of obtaining concomitant of system from concomitant of single qnantio 212 Complete list of covariants of system .... 218 Plane geometrical illustration of system of two cubios • . . 214 System 07 four oubics . . . . . ' 21ft Illustration of twisted cubics ..... 216 System of quartic aitd cubic ..... 218, 226 System of two quartics ..... 219 Their resultant ...... 220 Condition that u + \v should be perfect square . . . 220 Condition that « + Xv should have cubic factor . . . 221 Special form when both quartics are sums of two fourth powers . . 228 Tliree quadrics derived functions of a single quartic . . . 224 Three quadrics quadric covariants of two cubics . • . 22ft LESSON XVUL applications to higher binary quantics. Thbquintic ...... Canonical form of quintic .... Condition that two quartics be first differentials of the same quintic 227 227 228 XIV CONTENTS. Discriminant of qnintic ..... Fundamental invariants of qnintic Conditions for two pairs of equal roots All invariants of a quantic vanish if more than half its roots be all equal Hermite's canonical form ..... Hermite*s skew invariant . . . . Its geometrical meaning Covariants of quintic for canonical form Cayley's arrangement of these forms Cajley's canonical form Sign of discriminant of any quantic determines whether it has an odd or number of pairs of imaginary roots . . . Criteria furnished by Sturm's theorem for a quintio If roots all real, canonizant has imaginary factors Invariant expression of criteria for real roots Sylvester's criteria ..... Conditions involving variation within certain limits of a constant Cayley's modification of Sylvester's method . . Hermite's form^-type .... The Tflchimhausen transformation . - . Kodified by Hermite and Cayley Applied to quartic ..... Applied to quintic . . . • Sextie resolvent of a quintic Harley's and Cockle's resolvent Expression of invariants in terms of roots The sbxtio — ^its invariants and simplest covariants Conditions for cubic factor or for two square factors The discriminant .... Simplest quartic covariant .... Quadric covariants .... The skew invariant expressed in terms of other invariants Functions likely to afEord criteria for real roots . System of two quabtics .... Jacobian identified with any sextic by means of a quintic Functional determinant of three quartics . Can be similarly identified . . . New canonical forms of sextic by Brill Also by Stephanos .... Factors of discriminant of Jacobian . Sextic covariant of third order in coefficients Canonical form referred to ternary system Condition for sextic to be sextic-covariant of quartic to be Hessian of quintic LESSON XIX. ON thb order of restricted systems of equations. Order and weight of systems defined .... Restricted systems ...... Determinant systems, k rows, k + l colunms Order and weight of conditions that two equations have two common roots 229 232 23^ 233 233 233, 282 234 235 237 239 even 239 240 241 241 245 246 248 249 250 251 252 254 267 257 258 260 263 263 266 268 269 271 271 273 274 275 276 277 278 279 281 282 282 284 285 287 291 CONTENTS. X7 PAOM Sjvtem of conditioiLB that three equations shoold have a common root 293 Systems of conditions that equation have cubic factor or doable square factor 294 Intersection of quantics having common curves .... 296 Case of distinct common carves ..... 297 Number of qnadrics passing through five points and touching four planes . 298 Bank of curve represented by a system of k rows, k + 1 columns . 299 System of conditions that two equations should have three common roots . 800 System of quantics having a sorfaoe common . . . 801 Having common surface and curve .... 808 Having common two surfaces .... 806 System of conditions that three ternary quantics have two common points 806, 809 Bole when the constants in systems of equations are connected by relationa 807 Number of curve triplets having two common points . . 810 HCr. S. Boberts' method ...... 810 LESSON XX APFLI0ATION8 OF SYMBOLICAL METHODS. Symbolical expression for invariant or covariant 814 Clebsch's proof that every covariant can be so expressed . . 816 FormulsB of transformation .... 816 Beduction to standard forms .... . 818 Transvection ...... 820 Symbolical expression for derivative of derivative . 821 Forms of any order obtained by transvection from forms of lower ordisr 828 Gordan and Clebsch's proof that the number of irreducible covariants is finite 824 Every invariant symbol has {ab)P as a factor, where /? is at least half n 826 Symbolical expression for resultant of quadratic and any equation 826 Investigation of equation of inflezipnal tangents to cubic 880 Application of symbolical forms to theory of double tangents to plane carves . 888 laical exposition of an even binary quantic 886 of a quantic of order 3p . 887 NOTES. History of determinants ..... 98S Ck)mmutants ..... r 889 On rational functional determinants 840 Hessians* ...... . 841 Symmetric functioDS . * . . . 842 Elimination ....*. 842 Discriminants ..... 842 Beisoutiants ...... 843 Linear transformations ..... 843 Canonical forms . . . . . 846 Ck)mbinants ...... 846 Applications to binary quantics .... 846 Table of transvectants ..... 846 H. Boberts' table of sums of powers of difEeiences 347 Table of resultants ..... 848 Hirsch and Cayley's tables of symmetric functions 860 Index ...... 357 Errata* 40, line 1 after first determinant, in80ri " conjugate " before " imaginary." 63, Art. 62 line 4, and p. 64 line 2, write/or X, Xy : also in third linefollowingybr A, Xx, 136, last sentence of Art. 148^ interchange " order in the coefficients" with " degree in the Yariables/* 139, line 4 from end of Art. 163, read "coefficient ", 196, line 12, add after " thus " " the sqnares of the factors of 27 are the YalneB of " : also in Ex. 3 and p. 201 Ex. 2 read " the squares of the faptors/' 216, line 6, after JT, pat a semicolon. 226, Ex. 10, for Art. " 216" read " 219^." 236, line 10 from bottom, add " of T and U:* 243, line 2 begms, « *</>' (a) + «t/r (a)." 247, line 2, read "+ i /D.", as on p. 260. 269, Add as footnote to Art. 260, on £ as given in the second Edition, '^ where the following corrections should be made p. 269, col. 3 line 6, coefficient - 96 should be + 24, p. 266 „ 3 „ 13, „ -360 „ „ +20300." 279, Ex. 3, at end o<W « of Art. 266." 279, Ex. 4, the third determinant form for/ should have — before it. 280, Ex. 6, line 2, read "(Art. 267)." 800, Art. 286, lines 1 and 8,/or Hha last article" read "Art. 272." LESSONS ON HIGHER ALGEBEA. LESSON I. nBTEBBONANTS.— PRELIMINARY ILLUSTRATIONS. 1. If we are given n homogeneoas eqaatioDS of the first degree between n variables, we can eliminate the variables, and obtain a result involving the coefficients onlj, which is called the determinant of those equations. We shall, in what follows, give rules for the formation of these determinants, and shall state some of their principal properties ; but we think that the general theory will be better understood if we first give illustrations of its application to the simplest examples. Let us commence, then, with two equations between two Tariables Adding the first equation multiplied bj i, to the second multiplied by - J^, we get a,J, — ajb^ = 0, the left-hand member of which is the determinant required. The ordinary notation for thb determinant 19 We shall, however, often, for brevity, write {afi^) to express this determinant, leaving the reader to supply the term with the negative sign; and in this notation it is obvious that [afi^ = — [ajb^). The coefficients a„ i„ &c,, which enter into the expression of a determinant, are called the constituents of that determinant, and the products a,&,, &c., are called the elements of the determinant. B DETERMINANTS. 2. It can be verified at once that we should have obtained the same result if we had eliminated the variables between the equations In other words «81 K or the value of the determinant is not altered if we write the horizontal rows vertically, and vice versd, 3. If we are given two homogeneous equations between three variables, these equations are sufficient to determine the mutual ratios of Xj y^ z. Thus, bj eliminating y and x alternately, we can express x and y in terms of z^ when we find In other words, a?, y, z are proportional respectively to («A)» (a^&J, {ajb^. Substituting these values in the original equationS| we obtain the identical relations «, K&«) + % i^A) + «8 i^A) = ^> K («A) + K («8^) + h («A) = ; relations which are verified at once by writing them at full length, as for instance «i («A - ^A) + «. («8^ - «t^) + «8 («A - «A) = ^• The notation «1) «8) «8 (where the number of columns is greater than the number of rows) is used to express the three determinants which can be obtained by suppressing in turn each one of the columns, viz. the three determinants of which we have been speaking, {ajb^^ K*i))(«A)- 4. Let us now proceed to a system of three equations a,aj + &,y + Cji5 = 0, a^x + b^-^c^z^O^ a^X'{-bj/-\- c^z = 0. Then, if we multiply the first by {ajf^^ the second by (agJJ, the third by (^A)} ^^d add, the coefficients of x and y will vanish FOUND BT LINEAR ELIMINATION. 3 in virtue of tbe identical relations of Art. 3, and the deter- minant required is or, writing at full length, ^^M - «M + V»*i - ^M + «s«A - «»^«*i- It may also be written in either of the forms «i ( V.) + «. (V,) + «» (*A)) *i (^'a^J + K {<^fi^ + K K«J- This determinant is expressed by the notation «.» ^, c. «.. J., c. «.. *.. «. though we shall often use for it the abbreviation {ajbji^^ It is useful to observe that KVt) = K*A)i tot (a,J,cJ » - [afiji;j. For, by analogy of notation, K*A) = ^2 (*A) + «i(*A)+^i (*A)) which is the same as {afijy^^ whilo KVf) =^i A^a) + ^1 A^i) + «»(*i^s)i which is the same as - (a,J,Ci)- 5. We should have obtained the same result of elimination If we had eliminated between the three equations For if, proceeding on the same system as before, we multiply the first equation by (i.c,), the second by [cj:^^ and the third by (^A)) ^^^ ^^^' ^t^° ^^ coefficients of y and z vanish, and the determinant is obtained in the form which, expanded, is found to be identical with {fljb^o^. Hence «i) K ^x «IJ «2» % ««) *8V«8 = K K K «s? h ^S ^i» ^a) ^8 or the determinant is not altered by writing the horizontal rows vertically, and vice verad] a property which will be proved to be true of every determinant. DETERMINANTS. 6. Using the notation «l) «8» «85 «4 K K) K \ ^1? ^«J ^«? ^4 to denote the system of determinants obtained by omitting in turn each one of the columns, these four determinants are connected bjr the relations «i («« V4) - «a {^M + «a («4 V2) - «4 K^O = ^J ^ [^M - ^ (««^^i) + K iP'M - *4 («A^«) = o» These relations may be either verified by actual expansion of the determinants, or else may be proved by a method analogous to that used in Art. 3. Take the three equations a^x + ajf 4- a^z + a^ = 0, c^x + c^ + c^z + c^ = 0. Then (as in Art. 5) we can eliminate y and z by multiplying the equations by (J^cJ, (c^aj, (^A), respectively, and adding, when we find (^^j^^j 3. ^. (^^j^^J ii? = 0. In like manner, multiplying by {J>^c^\ (^s^i)) K^i) respectively, we get KVJy + («AO*'^ = o. And in like manner, («8* a) ^ + («4* a) ^ = ^• Now, attending to the remarks about signs (Art. 4), these equations are equivalent to or a?, y, «, w are respectively proportional to (^AOj "" {^J^4pi)i {ajb^c^\ — {flfi^c^ ; substituting which values in the original equations we obtain the identities already written. 7. If now we have to eliminate between the four equations a^x + hjf + c^z + d^w = 0, a^x-\-h^-\-c^z + djuo^% a^x -^ bj^ + c^z -{- d^w == Oj a^x + hj/ + c^z 4- rf^tt; = 0, SiaN OF AN ELEMENT. 5 we have only to multiplj the first by {ajk^c^j the second by — («8Vi)> ^^^ ^^^ ^y (^4^A)j *^® fourth by — (a,J,Cj), and add, when the coefficients of Xj y^ z vanish identically, and the determinant is found to be or^ writing it at full length, and altering all signs, «A^8^4 - «, V,< + «« Vl^4 - ^.^M + «a*M - «8Vl^4 + a, V«^8 - «lV4^8 + «4*«^l^8 - «4*M + «A^4^8 " «A«l^i + «sVl^8 - «8*.^4^« + «4*A^t - «4 VA + «J ^4^. - «1 ^8^. + a A^a^i - « A^4^i + «4*8^«^i - «4* A^i + «a* A^i - «8 V A • 8. There is no difficulty in extending to any number of equa- tions the process here employed; and the reader will observe that the general expression for a determinant is 2 ± ^i^8^8^« &c., where each product must include all the varieties of the n letters and of the n suffixes, without repetition or omission, and the determinant' contains all the 1.2.3. ..n such products which can be formed. With regard to the sign to be affixed to each element of the determinant, the following is the rule : We give the sign + to the term afi^c^d^ &c., obtained by reading the determinant from the left-hand top to the right-hand bottom comer; and then "<Ae sign + or — w affixed to each other product according as it is derived from this leading term by an even or odd number of permutations of suffi^es.^^ Thus, in the last example, the second term afi^c^d^ differs from the first only by a permutation of the suffixes of b and c ; it therefore has an opposite sign. The third term, apf^d^^ differs from the second by a permutation of the suffixes of a and c; it therefore has an opposite sign to the second, but it has the same sign with the first term, since it can be derived from it by twice permuting suffixes. Ex. In the determinant ifl^^x^K^hi^ what sign is to be prefixed to the element From the first term, permuting the suffixes of a and c, we get (ij>^id^e^^ the first constituent of which is the same as that in the given term; next permuting the suffixes of b and e, we get ajtfp^d^e^^ which has two constituents the same as the given term ; next, permuting c and e, we get ajb^c^^e^ \ lastly, permuting d and e, we get the given term ajb^c^^e^. Since, then, there has been an even number (four) of permutations, the sign of the term is +. In fact, the signs of the series of terms are ttib^^d^e^ — a<^^id^^ + ajt^e^d^^ — a^b^Cjd^Ci + Otb^Cjdie^. 6 DETERMINANTS. ^H Tbe rule of signs may otherwise be presented thns: wfr*' take for each suffix so often as It cornea after a superior suffix the sign — , and compound these into a single sign + or — . TLu3 comparing tbe elements afi^'^gfit'^f,, "s^jCjrf.e,, it will be seen that the suffix 1 which came first in the former element, ia in the latter preceded by three constituents ; that tbe suffix 2 is preceded by two which came after it before, and the suffix 4 by one. The total number of displacementa is therefore six, and this being an even number, the sign of the term is positive. Thus the rule ia, that the sign of the term is positive when the total number of displacements, as compared with the order in the leading term, is even, and vice versd. The same results will be obtained if, writing the auffixes always in the order 1, 2, 3, &c., we permute the letters, giving to each arrange- ment of tbe letters its proper sign + or — according to tbe rule of signs. Thus tbe determinant of Art. 7 might be written 9. A cyclic interchange of suffixes alters the sign when the number of factors in the product is even, but not so when the number of factors is odd. Thus aji^, heing got from afi^ by one interchange of suffixea, has a different sign ; but «,&,c, has the same sign with a^b^c^, from which it is derived by a double permutation. For, cbanging the suffixes of a and S, o,i,c, becomes a,&,Cg, and changing Ibe suffixes of b and c, this again becomes aji^c^. lu like manner aj)^c^d^ has an opposite sign to aji^c^d^, being derived from It by a triple permutation, viz. through the atepa aji^c^d^, aJ>f^d^, o^ftd,- This rule enables us easily to write down tbe terms of a determinant with their proper signs, by taking the cyclic permutations of each arrangement. Thus, for three rows tbe arrangements of suffixes are evidently + 123, + 231, -f 312, and — 213, — 132, — 321. For four rows the arrangements are + 1234 - 2341 + 3412 - 4123 ; - 1243 + 2431 - 4312 + 3124 ; - 1324 + 3241 - 2413 H- 4132 ; + 1423 ~ 4231 + 2314 - 3142. 10. We are now in a position to replace our former definition of a determinant by another, which we make the foundation GENERAL DEFINITION. of the subsequent theory. In fact) since a detenninant is only a function of its constituents a^j b^j c^j &c., and does not contain the variables Xj y^ e, &Cj It is obviously preferable to give a definition which does not introduce any mention of equations between these quantities a?, y, z. *Let there be n' quantities arrayed in a square of n columns and n rows, then the sum with proper signs (as explained, Art. 8) of all possible products of n constituents, one constituent being taken from each column and each row, is called the determinant of these quantities, and is said to be of the n^ order. Con- stituents are said to be conjugate to each other, when the place which either occupies in the horizontal rows is the same as that which the other occupies in the vertical columns. A deter- minant is said to be symmetrical when the conjugate constituents are equal to each other ; for example, A, i, / 11. In these first lessons, as in the previous examples, we usually write all the constituents in the same row with the same letter, and those in the same column with the same suffii^. A common notation, however, is to write the constituents of a determinant with a double suffix, one suffix denoting the row and the other the column, to which the constituent belongs. Thus the determinant of the third order would be written or else ^1,1) ^1,8) \z «»,!» «»,«) «8,B «8,1) «8,«) «B,B h * We might have commenced with this definition of a determinant, the preceding articles being unnecessary to the scientific development of the theory. We have thought, however, that the illostrations there given would make the general theory more intelligible ; and also that the importance of the study of determinants would more clearly appear, when it had been shown that every elimination of the variables from a system of equations of the first degree, and every solution of such a system, gives rise to determinants, such systems of equations being of constant oocorrenoe in eveiy department of pure and applied mathematics. 8 DETERMINANTS. where, in the sum, the suffixes are interchanged in all possible ways. The preceding notation is occasionally modified by the omission of the letter a, and the determinant is written 11, 12, 21, 22, 31, 32, 33 13 23 (1,1), (1,2), (1,3) (2,1), (2,2), (2,3) or (3, 1), (3, 2), (3, 3) Again, Dr. Sylvester has suggested what he calls an umbral notation. Consider, for example, the determinant aa, 5a, ca, doL a/3, 6/3, c/3, rf/3 07, J7, C7, dy aSj b8j cS, dS the constituents of which are aa, 5a, &c., where a, 5, c^ &c., are not quantities, but, as it were, shadows of quantities ; that is to say, have no meaning separately, and only acquire one in combination with one of the other class of umbrae a, ^, 7, &c. Thus, for example, if a, yS, 7, S represent the suffixes 1, 2, 3, 4, the constituents in the notation we have ourselves employed are all formed by combining one of the letters a, 5, c, d with one of the figures 1, 2, 3, 4. Now the above determinant is written by Dn Sylvester more compactly a, a *, /3 c, 7 d, S which denotes the sum of all possible products of the form aa.5y9.C7.rfS, obtained by giving the terms in the second column every possible permutation, and changing sign according to the foregoing rule of signs. Observe that if the two columns are identical, and if in general rs means the same thing as sr^ then the determinant is symmetricaL ( 9 ) LESSON IL EEDUCnON AND OALOULATION OF DETEBHINANTS. 12. We have in the last Lesson given the rale for the forma- tion of determinants, and exemplified some of their properties in particular cases. We shall in this Lesson prove these pro- perties in general, together with some others, which are most fireqnentlj used in the reduction and calculation of determinants. The value of a determinant is not altered if the vertical columns he vnitten horizontally^ and vice versd (see Arts. 2, 5). This follows immediately from the law of formation (Art. 10), which is perfectly symmetrical with respect to the columns and rows. One of the principal advantages of the notation with double suffixes is that it exhibits most distinctly the symmetry which exists between the horizontal and vertical lines. 13. If any two rows [or two columns) he interchanged^ the sign of the determinant is altered. For the effect of the change is evidently a single permutation of two of the letters (or of two of the suffixes), which by the law of formation causes a change of sign.* 14. If two rows {or \f tioo columns) he identical^ the deter^ minant vanishes. For these two rows being interchanged, we ought (Art. 13) to have a change of sign, but the interchange of two identical lines can produce no change in the value of the determinant. * It may be remarked that a determinant is a fmiction which is determined (except for a common factor) by the properties that it is linear in respect of the constituents of each row and of each column, and that it merely changes sign if two rows or columns be interchanged. Thus for'two rows, the most general lineo-linear function of the rows and columns is and the condition that it is to change sign when we interchange a^ and b^ a^ and b^f gives -4 = 1) = 0, B+C= 0. The function is therefore C (a,6, — Oj*,), and if wo agree that the coefficient of afi^ is to be unityi the function is Oib^ — a^i as before. C 10 REDUCTION AND CALCULATION OF DETERMINANTS. Its value, then, does not alter when Its sign is changed ; that is to say, it is = 0. This theorem also follows immediately from the definition of a determinant, as the result of elimination between n linear equations. For that elimination is performed by solving for the variables from n — 1 of the equations, and substituting the values so found in the w'\ But if this w** equation be the same as one of the others, it must vanish identically when these values are substituted in it. 15. If every constituent in any row [or in any column) he multiplied hy the same factor j then the determinant is multiplied hy that factor. This follows at once from the fact that eveiy term in the expansion of the determinant contains, as a factor, one, and but one, constituent belonging to the same row or to the same column* Thus, for example, since every element of the determinant «1> K <*, %1 K <>» «8' K c. contains either a,, a^, or ag, the determinant can be written in the form a^A^-^- a^A^-^- a^A^^ (where neither -4„ A^^ nor -4, contains any constituent from the a column) ; and if a^, a^, a, be each multiplied by the same factor k^ the determinant will be multiplied by that factor. Cor. If the constituents in one row or column differ only by a constant multiplier from those in another row or column, the determinant vanishes. Thus = Jfc «87 «2J ^8 ^«? ^2) ^8 = (Art. 14). 16. If in any determinant we erase any number of rows and the same number of columns, the determinant formed with the remaining rows and columns is called a minor of the given determinant. The minors formed by erasing one row and one MINOR DETEUMIXANTS. 11 column may be called first minors; those formed by erasing two rows and two columns, second minors, and so on. We have, in the last article, observed that if the constituents of one column of a determinant be a^, a,, a,, &c., the deter- minant may be written in the form a^A^-^-a^A^-fa^A^+Scc. And it is evident that A^ is the minor obtained by erasing the line and column which contain a^, &c. For every element of the determinant which contains a, can contain no other con- stituent from the colunm a or the line (1); and a, must be multiplied by all possible combinations of products of n — 1 constituents, taken one from each of the other rows and columns. But the aggregate of these form the minor A^. Compare Art. 7. In like manner the determinant may be written a^A^ + b^B^^j^c^G^ + &c., where 5, is the minor formed by erasing the row and column which contain b^. 17. If all the constituents but one vanish in any row or column of a determinant of the n^^ order ^ its calculation is reduced to the calculation of a determinant of the n — 1*** order. For, evidently, if a^, ttg, &c., all vanish, the determinant a^A^ + a^A^ + &c., reduces to the single term a^A^ ; and A^ is a determinant having one row and one column less than the given determinant. Conversely, a determinant of the n — 1*** order may be written as one of the n^^ or higher order. Thus &c. b • c 1, 0, 1| K c, *8. ^3 = 0, h, c. or = 0, h, c. 0, *., c. 0, K 0. 18. If every constituent in any row {or in any column) be resolvable into the sum of two others^ the determinant is resolvable into the sum of two others. This follows from the principle used in Art. 16. Thus, if in the Example there given, we write a^ + a, for a^\ b^-\- /8, for b^ ; Cj + 7j for Cj ; then the determinant becomes (a. + a.) ^. + (^ + ^.) A + (c. + y.) ^» = K^. + *. A + c. G,] + {a,A, + ^.5, + 7. (7.}. 12 REDUCTION AND CALCULATION OF DETERMINANTS. Thus we have ^1) ^a) ^8 «1) «2) «8 7„ <5,) <58 K + A) ^» ^ Ci+7i, c„ C3 In like manner, if the constituents in any one column were each the sum of anj number of others, the determinant could be resolved into the same number of others. 19. If again, in the preceding, the constituents in the second column were also each the sum of others (if, for instance, we were to write for a,, a, + a^ ; for &j, &, + yS, ; for c,, c, + 7,), then each of the determinants on the right-hand side of the last equation could be resolved into the sum of others ; and we see, without difficulty, that And if each of the constituents in the first column could be resolved into the sum of m others, and each of those of the second into the sum of n others, then the determinant could be resolved into the sum of mn others. For we should first, as in the last Article, resolve the determinant into the sum of m others, by taking, instead of the first column, each one of the 7n partial columns ; and then, in like manner, resolve each of these into n others, by dealing similarly with the second column. And so, in general, if each of the constituents of a determinant consist of the sum of a number of terms, so that each of the columns can be resolved into the sum of a number of partial columns (the first into m partial columns, the second into n, the third into p, &c.), then the determinant is equal to the sum of all the deter- minants which can be formed by taking, instead of each column, one of its partial columns ; and the number of such determinants will be the product of the numbers 9n, n,p, &c. 20. If the constituents of one row or column are respectively equal to the sum of the corresponding constituents of other rows or columnsj multiplied respectively hy constant factors^ the deter^ minant vanishes. For in this case the determinant can be resolved into the sum of others which separately vanish. CALCULATION OF NUMERICAL DETERMINANTS. 13 Thus Bat the last two determinants vanish (Cor., Art. 15). ^81 ^f1 ^8 21. ^ determinant is not altered if we add to each constituent of any row or column the corresponding constituents of any of the other rows or columns multiplied respectively by constant factors. Thus «i) «8» «8 ^j *8» K C,) ^81 ^. But the last determinant vanishes (Art. 20).* The following examples will shew how the principles just explained are applied to simplify the calculation of determinants. Ex. 1. Let it be required to calcnlate the following determinant : 9, 13, 17, 4 18, 28, 83, 8 30, 40, 54, 13 24, 37, 46, 11 1, 1, 1, 4 1, 1, 1, 1 2, 4, 1, 8 ^ 2, 4, 1, 1 4, 1, 2, 13 4, 1, 2, 6 2, 4, 2, 11 2, 4, 2, 8 The second determinant is derived from the first by subtracting from the constituents of the first, second, and third columns, twice, three times, and four times, the corre- sponding constituents of the last column. The third determinant is derived from the second by subtracting the sum of the first three columns from the last. When- ever we have, as now, a determinant for which all the constituents of one row are equal, we can get by subtraction one for which all the constituents but one of one TOW vanish, and so reduce the calculation to that of a determinant of lower order (Art. 17). Thus subtracting the first column from each of those following, the deter- minant last written becomes 1, 0, 0. 2, 2,-1,-1 4,-3,-2, 2 2, 2, 0, 1 The third of these follows from the second by subtracting twice the last column from the first, leaving a determinant of only the second order, whose value is - 8 - 7 = - 15. 2,-1,-1 4,-1,-1 4,-1 -8,-2, 2 — -7,-2, 2 = -7,-2 2, 0, 1 0, 0, 1 * The beginner will be careful to observe that though the determinant is not altered if we substitute in the first row a^ + ka^ + la^ for aj, dec, yet if we make the same, substitution in the second row for a,, (&c., we multiply the determinant by k ; and if in the third for ag, dec, we multiply it by U 14 REDUCTION AND CALCULATION OP DETERMINANTS. K.B. — It is in all cases possible to make all actual constituents in any row of the same valae — by multiplying each and the terms in its colamn by the product of the others. Li this way a determinant can, as in this example, be reduced to one of lower order, and this process is generally the best for numerical calculation. Ex. 2. The calculation of the following is necessary {Surfaces, Art. 234) : 5, - 10, 11 - 32, - 35, 34 1, 6, 8 = 10 6, - 10, ] 11, 5, - 10, 11 ,0 - 10, - 11, 12, 4 11, 12, - 11, 2 = - 32, - 35, 34, 11, 12, - 11, 2 = -2 0, 4, 2,-6 1, 6, 8, 6,-2,1 5,-2, 1 • 9 11 32, 7, 1 1, 1,8 = 10 27, 9, - 39, 17, = 90 Of X - 39, 17 = 90 (51 +39) = 8100. The first transformation is made by subtracting double the third row from the second, and adding the sum of the second and third to the fourth. In the next step it will be observed, that shice the sign of the term a^^^d^ is opposite to that of a^^zd^ when C4 is the only constituent of the last column which does not vanish, the deter- minant becomes — c^ {P'^%d^» In the next step, we add the second and third columns, we take out the factor 5 common to the second column, and the sign — common to the second row. We then subtract the first row from the second, and eight times the first row from the last, and the remainder is obvious. Ex. 8. Ex.4. 7,-2, 0,5 -2, 6,-2,2 0, - 2, 5, 3 6, 2, 3, 4 25 , - 15, 23,-6 -15 , - 10, 19, 5 23; , 19, - 15, 9 - 6 , 5, 9,-5 = - 972 {Surfacesj Art. 226). = 194400 {Surfaces, Art. 234). Ex. 5. Given n quantities a, /3, y, &c., to find the value of 1, 1, 1, 1, &c. «» A y» ^, Ac. ^, y^ ^*, Ac. tt»-l^ ^-1^ y«-l^ ^»-»,*^&C. It is evident (Art. 14) that this determinant would vanish if a = /3, therefore a — /3 is a factor in it. Li Uke manner so is every other difference between any two of the quantities a, /3, (&c. The determinant is therefore = 03 - a) (y - a) (^ - «)...(y - ^) (« - /3)...(« - y)...&c. For the determinant is either equal to this product or to the prx)duct multiplied by some factor. But there can be no factor containing o, /3, Ac, since the product con- tains a"-*, /3»-*, (fee. J and the determinant can contain no higher power of a, /3, Ac. ; and by comparing the coefficients of a*-* it will be seen that the determinant contains no numerical factor. This example may also be treated in the same way as the next example. EXAMPLES. 15 Ex. 6. To calculate 1, 1, 1, «» fit y» 1 d ««, P», y\ ^ a\ /3*, y*, a* Sabtract the last column from each of the first three and the determinant becomes divisible by {6 — o) (d — /3) {6 — y), the quotient being 1, 1, 1 o + a, /3 + a, y + d o« + o2a + oa» + a», /8» + /3«a + /33« + a», y» + y«a + y3« + a» Sabtract again the last colnmn from the two preceding and the determinant is seen to be divisible by (a — y) (Ji — y), and its valae la thos at once found to be (a-a)(i3-a)(y-a)(a-y)(/3-y)(a-/3)(a + /3 + y + a).* Ex. 7. In the solution of a geometrical problem it became necessary to determine X from the equation (a + X)» (6 + X)», (c + X)« (2a + X)», (2* + X)», (2tf + X)» = 0. Subtract the first row from the second^ and divide by X ; subtract 8 times the first row from the last and divide by X ; then subtract the second row from the third and divide by 8 ; and, lastly, subtract this last row from the second and divide by X, when the determinant becomes 2a + X, 2^+ A, 2c + X 8o» + aX, S^ + b\ B<^ + c\ = 0. Again, subtract the first column from the second and third, and divide by b — a, e — a; then subtract the second from the third, and divide hj c — b; and then from the first column subtract a times the second and add o^ times the last ; and from the second column take (a + b) times the last, and we have finally tibCf — (fl5 + 6c+'ca), a + b + e X, 2 0, X, 8 = 0, which reduced is {a + b + c)\^ + B {ab + bc + ea)\ + Sabc = 0. Ex.8. {b + c)«, a«, a« i», {e + a)«, *« c», <?«, (a + 6)« = 2abc (a + ft + <?)«. Ex.9. 1, 1, 1 sin a, 8in/3, siny cos a, C08/3, cosy = 4 sin J (a - /3) sin J (/3 - y) sini (a - y). * On the general theory of which this and the preceding example form part, see Jiote at end on Bational Functional Beterminanta. 16 REDUCTION AND CALCULATION OF DETERMINANTS, Similarly, by Ex. 6, Bin'o, Bin'a cos a, sin a oos'a, cos'o, an*Py sin^^cosjg, sin/3cos2/3, cos^/S, sin'y, sin^ycosy, sinyoos'y, cos'y, sin^d, sin'd cos d, sin d cos^d, cos'd, sin (a — /3) sin (a — y) sin (a — d) sin (/3 — y) sin (/3 — d) sin (y — d). Ex. 10. cos (o — /3), cos f)3 — y), cos (y — a) cos (o + /3), cos (/3 + y), cos (y + o) sin (a + /3), wa.(fi + y), sin (y + o) = 2 sin (o — /3) sin (/3 - y) sin (o— y). Ez.lL 1, a«, *« a, ft, oft 1, a^ ft'' -2 a', ft', a'ft' 1, a"2, *"« a", ft", a"ft' sin a, cos a, sin a cos a sin/3, cos/3, sin/3 cos/3 siny, cosy, siny cosy = 2 sin J (/3 - y) sin i (y - «) sin i (a - /3) {Bin (o + /3) + sin (/3 + y) + sin (y + o)}. This follows at once from the identities (a'ft" + a"ft' + o"ft + oft" + oft' + a'ft) 1, a, ft 1, a', ft' 1, o", ft" o'ft" + a"ft' - 2aft, a, ft o"ft + aft" - 2a'ft', o', ft' aft' + a'ft - 2a"ft" a", ft" Ex. 12. Many of these examples may be applied to the calcnlation of areas of triangles, it being remembered that the doable area of the triangle formed by three points is 1, 1, 1 y', y", y" and by three lines ax+by + Cj dec, is a, b, c ^ a', ft', c* a", ft", c" divided by (a'ft" - a"ft') (o"ft - aft") (aft' - a'ft) (see Cbntc Sections f p. 32). For example, the area of the triangle formed by the centres of cnrvature of three points on a parabola is (the coordinates of a centre of curvature being ^p + 8a;, — ^) tt III 6 =^(y'-y")(y"-y)(y-yO(y'y"+y"y+yyO. f 1, 1, 1 y\ y*, y"* y», y", y"» In like manner may be investigated the area of the triangle formed by three normals, or any other three lines connected with the curve. Ex.13. 0, c, ft 0, <?, ft d 0, 0, a <;, 0, a, e ft, a, -2aftc; *, «, 0, / <;, e, /, — /»2, c^fft V + <j»/2-2aft(fe - 2ft<?e/-2a<fcy . EXAMPLES. 17 Ex. R Proye 0, 1, 1, 1 1, 0, ««, y« 1, «« 0, ^ 1, y^, ^y 0, a?, y, e X, 0, e, y y, Zy 0, « «, y, it, and I, jc, y, « it, ty «, y y, «, «, aj «, y, it, % = (< + it + y + «)(< + a> - y -«)(*- a? + y -«)(<- iB - y + «). Ex. 15. a, X, X, X, &o. X, 5, X, X, d(0. A, X, Cy X, (&0. X, X, X, dy ^0. Ac. where all the oonstittients are equal except those in the principal diagonal, ia 4> (X) — X ^; where ^ (X) is the oontinned product (a — X) (6 — X) Ac Ex. 16. Let « be a homogeneons function of the it*'* order in any number of Tariables; and let u-^y u^ u^ (&o., denote its differential coefficients with regard to the variables x^y x^ ar„ &c. ; and in like manner, let Uu, u^^ Ui, denote the difEer- ential coefficients of «,, dec. Then, bj Euler's theorem of homogeneous functi(Hi8» we have »« = ttjfl?, + «»», + t<,a?8 + 4c., (n - 1) tt, = af,tii, + iB,«i, + itjt«„ + *c., Ac We shall hereafter speak at length of the determinant (called the Hessian) formed with the second differential coefficients, whose rows are v,,, u^^ u^si Ac.; «2i) ^hta ^hxt 4c., Ac. At present our object is to shew how to reduce a class of determinants of frequent occurrence, which are formed by bordering the Hessian^ either with the first differential coefficients, or with other quantities, as for example «i«. tf «• •llJ «W* »M» "l» "I «2l, "22, «M, «» a» «M» ««» «»» «S» 0» «i, ttj, «„ 0, a„ Oj, a, 0, 0, In this example we only take three variables, and the determinant formed by the first three rows and columns is the Hessian, which we shall call /T, but the processes which we shall employ are applicable to the case of any number of variables. We denote the above determinant by the abbreviation [ j , and use f j , [ j , [ \ to denote the determinants of four rows, formed by bordering the Hessian with a single row and column, either both u's or both a's, or one u and the other a. We also write a^x^ + a^a^ + a^, = a. If now we multiply the first column of the above written determinant by X|, the second by x^ the third by x^ and subtract the sum from n — 1 times the fourth column, the first three terms vanish, the fourth becomes — nuy and the fifth — a. Again, multiply the fourth row by n ~ 1, and subtract in like manner the sum of the first, second, and third rows multiplied by x-^y cn^ Xg respectively, then the first three terms vanish, the fourth remains unchanged, and the last becomes - a. Thus then (n — 1)' times the determinant originally 18 EXAMPLES. written 18 proTed to be equal to **11» **!» *'l» *4i> ^hsf **«» **«» **!» *^» 0, 0, 0, «i» <h) 'hf 0, 0, 0, — « (n — 1) fi, — o — o. But now since (Art. 15) a determinant which has only two terms d^j d^ of the fourth row which do not vanish, is expressible in the form d^B^ + d^L^ ; the above deter- minant may be resolved into the sum of two others, and we find that the originally given determinant \u aj » — 1 \aj (» — \y In like manner it is proved that ( ) = -^ Hu, Or, again, if there be four variables, and the Hessian be triply bordered, we prove in the same way that c:?)=-^«(:^(^.-'©-(^^(:)-(;rii).'^©- When « is of the second degree, it is to be noted that, in the case of three variables, [ j = is the condition that the line a should touch the conic u \ ^d f o] ^ ^ ^ the condition that the intersection of the lines a, /3 should be on the conic. In like manner for four variables the vanishing off J,f o)»( o) respectively, are the conditions that a plane should touch, that the intersection of two planes should touch, and that the intersection of three planes should be on, the quadric u. The equation then ( ** fl) ~ ^ expresses that the polar plane of a point passes through one or other of the two points where the line a/3 meets the quadric. But points having thifl property lie only on the tangent planes at these two points. The transformation, therefore, that we have given for f ^ j expresses the equation of the tangent planes at the points where aj3 meets the quadric, and the transformation f or [ " ] gives the equation of the tangent cone where a meets the quadric Ex. 17. Find the value of a determinant of the form a, 1, 0, 0, -1, ft, 1, 0, 0,- - 1, ^'i h 0, 0, - 1, c?, 1 0, 0, 0, - J, e Determinants in which all the constituents vanish except those in the principal diagonal and the two bordering minor diagonals, have been studied by Mr. Mnir under the name of continuants {Proceedings of the Royal 8oc.y Edinb,j 1878—4). The above determinant may be written in the abbreviated form (a, 5, c, dj e) ; and taking out the constituents in the first row (as in Art. 16), the value of the determinant is seen to be a (6, c, <?, e) + (c, <?, e). In this way we can easily form the series of values of continuants of two, thi-ee, &c, rows, viz. oft + 1, aibc + c + a, abed + {?ef + arf + oft + 1, abcde + dbe + ade + ade + cde + a + c + e, &c. MULTIPLICATION OF DETERMINANTS. 19 The rale of formation is, take the product abode of all the oonstitaenta, and omit from it in eyeiy possible way the pairs of consecutiye literal constituents. Thus, in the last case, the omitted pairs are de^ cd^ be, ah, (6c, de), (a6, de), (a6, cd). Determinants of the class here described occur in the theory of continued frac- tions; for it is obvious that the sucoesslye approzimations to the yalue of the oontinned fraction a + v— — - , Ac, are (a, 6) (fl, ft, c) («, h, c, rf ) Ex. 18. Find the number of terms in a continuant of the n'*^ order. From the equation (a, 6, <;, <^ e) = a (6, c, J, e) + (c, d^ e), it is obvious that if (n) be the number xequired, we have the relation (n) = (n — 1) + (n — 2) ; and that therefore for the orders 1, 2, 3, Ac, we have the series of numbers 1, 2, 3, 5, 8, 13, ^c.; and genenUy if /i±j(^Y = ^. + -B. J{6), the number required is il, + -Bifc. LESSON III. MULTIPLIOATION OF DBTERMINANTS. 22. The product of two determinants may be at once written down as a determinant whose order is the sum of their orders. For instance, the product of (a^i,), [^fi^^ naay be written down a 1) a„ , h, 0, 0, 0, a,, 0,, 0, 0, 0, p,, ^„ ^3 ^> 0> 7ii 7», 7. . 88 is evident on expanding the last written determinant. (See also post Art 32). We shall now shew that the product of two determinants may be written down as a determinant whose order does not exceed the highest of the two. The product of two determinants of the same order is the determinant whose constituents are the sums of the products of the constituents in any row of one by the corresponding constituents in any row of the other. . 20 MULTIPLICATION OF DETERMINANTS, For example, the product of the determioants {afi^c^ and a,a, + 5^, + c,7„ o,a, + J,/9, + c,7„ a A + 5,/9, + 0,73 The proofs which we shall give for this particular case will apply equally in general. Since the constituents of the deter^ minant just written are each the sum of three terms, the de- terminant can (by Art. 19) be resolved into the sum of the 27 determinants, obtained by taking any one partial column of the first, second, and third columns. We need not write down the whole 27, but give two or three specimen terms ; ^1«J7 ^l°^2» ^l°^8 a^aj, a^a,, a^a^ ^8°^l) «8^8) «8«S ^i^i) ^i7«) *i^t ^a^i, Cj78) *A ^^ij ^8787 *»^a + ^ «1) «1) ^J «i» ^) ^1 «i) ^1) ^ a„ Oj, a. + ai^«78 a„ h^y c. + «i7 A ««» ««) *» «8» ^81 <^8 «8? *8» ^8 S» S» *8 + «,«„ *Aj ^278 Now it will be observed that in all these determinants each column has a common factor, which (Art. 15) may be taken out ,. as a multiplier of the entire determinant. The specimen terma already given may therefore be written in the form *^v^J^ But the first of these determinants vanishes, since two columns are the same ; the second is the determinant {ajb^c^ ; and the third (Art, 13) is = — (^j^gCg). In like manner, every other partial determinant will vanish which has two columns the same ; and it will be found that every determinant which does not vanish will be [afi^c^^ while the factors which multiply it will be the elements of the determinant {oL^0^y^)» It would have been equally possible to break up the deter* minant into a series of terms, every one of which would have been the determinant {oLi^^y^) multiplied by one of the elements The theorem of this article is applicable to the multiplicatioii of determinants of different orders, because it has been shewn (Art. 17) that we can always write that of the lower order as one of the same order with the higher. ANOTHER PROOF. 21 23. On accooDt of the importance of this theorem^ we give another proof, founded on our first definition of a determinant. The determinant which we examined in the last Article is the result of elimination between the equations (a,a, + JA + C97i)aJ+ (a,a,+ J,/9,+ c,7,)y + (a,a,+ J^,+ c,7,)«=0, K«i + ^'Si + C37,)a:+(a3a,+ J,/93+ c,7jy + (a,a,+ J3/93+ 0,7,) «=0. Now if we write 7i« + 7>y + 78« = ^> the three preceding equations may be written from which, eliminating X, Y, Z, we see at once that {dfi^c^ must be a factor in the result. But also a system of values of a?, y, z can be found to satisfy the three given equations, provided a system can be found to satisfy simultaneously the equations X=0, F=0, Z=0. Hence (0^/8,73) = 0, which is the con- dition that the latter should be possible, is also a factor in the result. And since we can see without difficulty that the degree of the result in the coefficients is exactly the same as that of the product of these quantities, the result is [afij^^ («i/^ii7t)* So in general the product of any number of determinants may be expressed as a determinant whose order does not exceed the highest of the orders of the given determinants, and whose elements are rational functions of the elements of the given determinants. It appears from the present Article that the theorem con* cerning the multiplication of determinants can be expressed in the following form, in which we shall frequently employ it: If a system of equations ajX+&,r+c,Z=:0, 03X4-637+ 03^=0, a^X-^-h^Y + c^Z^O 22 MULTIPLICATION OF DETERMINANTS. be transformed hy the substitutions then the determinant of the transformed system will be equal to (afi^c^) the determinant of the original system^ multiplied by (a,i8j7g), which we shall call the modulus of transformation. 24. The theorems of the last Articles may be extended as follows : We might have two sets of coustituents, the number of rows being different from the number of columns; for example «8» hy ^8 and from these we could form, in the same manner as in the last Articles, the determinant «A + *A+^i78) «A + &A + ^ii'y« whose value we purpose to investigate. Now, first, let the number of columns be greater than the number of rows, as in the example just written, so that each constituent of the new determinant is the sum of a number of terms greater than the number of rows ; then proceeding, as in Art. 22, the value of the determinant is a,a„ b^, + &c. = (^^s) (/3,7,) + (c.aj (%«,) + («A) («A)- That is to say, the new determinant is the sum of the products of every possible determinant which can be formed out of the one set of constituents by the corresponding determinant formed out of the other set of constituents, 25. But in the second place, let the number of rows exceed the number of columns. Thus, from the two sets of con- stituents, «1J *. «i> /3, ««>*. «../3. «•> h «•) ^» EXAMPLES* 23 let us form the determinant «,«! + * A, aji^-¥hfi,^ «a«i + *A ^A + ^^8l «8«a + *A» ^3«8 + ^a^a Then when we proceed to break this up into partial determinants in the manner ah^eady explained, it will be found impossible to form any partial determinant which shall not have two columns the same. The determinant^ therefore^ will vanish identically. Or this may be seen immediately by adding a column of cyphers to each matrix and then multiplying, when we get the determinant last written as the product of two factors each equal zero. 26. A useful particular case of Art. 22 is, that the square of a determinant is a symmetrical determinant (see Art. 10). Thus the square of {afi^c^) is Again, it appears by Art. 24, that the sum of the squares of the determinants (i,c,)' 4- (c^a,)' + (a,6J' is the determinant a,a, + i^J, + c,c„ a* +5,* +c/ Ex. 1. If Oif bif C| ; o^ 5^ c, be the direotion-oosines of two lines in space, and their inclination to each other, cos = ai<i^ + bib^ + «|C^ ; and the identity last piOTed gives 8in*0 = {biC^y + {e^a^* + (ajA,)*, Ex. 2. In the theoiy of eqnations it is important to express the product of the Bqnares of the differences of the roots; now the product of the differences of n quantities has been expressed as a determinant (Ex. 5, p. 14), and if we form the equaie of this determinant we obtain 'i» Hi '• ••• '» where fp denotes the sum of the^^ powers of the quantities a, p, ibc Ex. 3. In like manner it is proved, by Art. 21, that the determinant '0» 'l> *2 = 2:03-y)«(y-a)«(a-^)*. 24 EXAMPLES. We thus fona a series of determinants, the last of which is the product of the squares of the differences of a, /3, dec. ; all similar determinants beyond this yanish identicallj by Art. 25. This series of determinants is of great importance in the theory of algebraic equations. Ex. 4. If there be any n quantities a, )9, y, dec., and a Uke number a\ /S', y% &Q.f the product may be formed o", ro'->, Jr (r - 1) o'*...! l-Zr, /S'2...(-^')r (a - a'Y, (a - a^, (a - y'Y ... (^-a'r, (^-^OSOS-y'r- when if the number n > r + 1 each side yanishes \ otherwise as each determimmt of the first matrix contains as factors every difference of the first quantities, and each of the second every difference of the second set, the resulting determinant is divisible by the product of both continued products of differences. Ex. 5. Various well-known identities may be established by the method of Art. 25. Thus It is easily seen that cOs a, sm a cos/3, sin/3 cosy, siny cos a, sm a cos^ Bin/3 cosy, siny gives similarly 1, cos 03 — a), cos (y — o) cos (a — /3), 1, cos (y — /3^ cos (a — y), cos (/3 — y), 1 = 0, cos a, sma C06/3, sin/3 cosy, siny sin a', cos a* sin/y, cos/y siny', cosy' = 0. sin (o + a!), sin (P 4- a!), sin (y + a*) sin(a+/30, sin(/3 + /3'), sin(y+/3') sin(o + y*), sin(|3 + y'), sin(y + Y) If a conic break up into two lines {ax + /3y + ya), {a*x + /3'y + y'^), we find^ equating coefficients, 2a, 2A, 2^ 2A, 2ft, 2/ 2^, 2/, 2e a, a' a', a A^ ^', ^ = y» y' y'» y = (cf. Conies, Art. 77). If a biquadratic be of the form (aap + /3)* + {a'x + /S^*, by similarly identifying = 0. Ex. 6. In like manner, from Art. 24, the two methods of writing 02, o'« o2, o** a, ft, «A a'^ «A flt'^' = ft, c, d /8«, ^ ^, /3'2 <7, dy e a, ft, <; a', ft', c' c — 2ft, a c' - 2ft', a' 2(ac-ft2) ac' + a'c-2ftft' ac' + a'c - 2ftft', 2 (a'<?' - ft**) = 2 (ftc') (aft') - (cay + 2 (aft') (ftc'), give the identity 4 {ac - ft2) (aV - ft^^) - (ac' + o'(? - 2ftft')« = 4 (be') (aft') - (cay. EXAMPLES. 25 Again, (if bf Cf d a\ y, (f^ d* o« + i* + c' + cP, aa!'\-W +0^^+ <W = (&>')« +(<?a')*+(a*0* but (AcO (a<?') + {.car) (A<f) + (o^) (c<f) = = 0, a«f — o'rf, a, a' W - A'cf, A, y hence the right side may be written and the identity exhibits the product (a« + *» + ^ + <P) (a'» + ft** + tf^ + <?^, as the sum of four squaies, a theorem doe to Eoler. Ex. 7. The relation among six diBtances from a point along a line to points which form an involntion is that the determinant 1, Oj + a,, Ojo^ 1, ^1 + Jj, ii^i 1| Ci + c«, Cyp^ may yanish. (C9itic#, Note, p. 310). a*, — aj^ 1 Multiplying this by y', -^yi 1 %\ - «, 1 (a:-ai)(a:-o,), (y - Oi) (y - Oa), («-Oi) («-««) (x- h^) (»- 6J, (y- 6») (y- y, (»- ^) («- W (a:- <?i) (»- c,), (y- <?i) (y- <?t)» («- ci) («-«») by taking the yalues Oj for jb, 6| for y, ei for e, this reduces to 0, (*i - Oi) (*i - o,), (ci - a,) (cj - a,) (a, - ^i) (ai - 6j), 0, (<?, - fti) (Ci - *^ = (*i - Ci) (^'i - «i) («! - *i) {(ffi - *i) (*i - Ca) (ci - Oa) + (a, - ftj) (&, - <Ji) (<;, - aOlf whence the original determinant is expressed as the sum of the last two products^ or as 0, tf, - ij, c, ~ a, r, - aj, fti - Ci, Ex. 8. Let the origin be taken at the centre of the circle circumscribing a triangle^ whose radius is 22 ; and let M be the area of the triangle, then we get 2jrR = a^, y', R ^\ y", i2 and - 7MR = »', y*, -12 a^'. y", - i2 Multiply these determinants according to the rule, and the first term a/' + y^ — 12* vanishes; the second aV + y'y" - 12» = - J {(«' - «")* + (y' - y")*} = - ic«, wher» c is a fdde of the triangle. Hence then 0, c2, 6« c«, 0, a* 6«, a«, whence 12 = tt>} ^ ^ ^^^ known. 4Jf - 4Jf «12« = - i = - Ja«AV, 26 EXAMPLES* Ex. 9. The same process may be used to find an expression for the radius of the sphere circnmscribing a tetrahedron. Starting with the expression for the Tolwne of tlie tetrahedron 6F = z J' ji 1 a", y ', «", 1 a^", y"\ x"\ 1 a; /K /K .'K We find| as before, ita, d; d, e ; Cf/axe pairs of opposite edges of the tetrahedron 0, C2, i«, CP ^y e^ P. whence if a</ + ie + 0/"= 25; by Ex. 13, p. 16, we find Z^B^V^ = 8 {8 -ad) {8 -be) {8 - ef), Ex. 10. The above proofs by Mi*. Burnside were suggested by the following proof by Joachimsthal of an expression for the area of a triangle inscribed in an ellipse. Multiply the equations ^ y' 1 a» b' ^ x' y' a' b' -1 2M ab " ' lib~ x" y" a' 1' -1 x'" y'" . x"' y'" -1 And the product is a symmetrical determinant, of which the leading terms, such as -J- + ^ — 1, vanish when the points a^y', x"y^', x"'y'" are on the curve, while the other terms are — s- + -?^ — 1, Ac. Now it can easily be proved that if y be a Bzde of the triangle, and V" the parallel semi-diameter, b"'2' o2 "*" 62 ~ V o* 6«y* Thus we have o«4« * ^ E. /3^ a2/32y5 4d'2d"*6'"« ' Ex. 11. The following investigation of the relation connecting the mutual distances of four points on a circle (or five points on a sphere) was given by Prof. Cayley {Cam' bridge and Dublin Mathematical Joumaly vol. ii., p. 270). Substitute the coordinates of each point in the general equation of a circle aj2 + y2 - ^Ax - 2By + C = 0, and eliminate A^ B, C, when we get a determinant with four rows such as x'^ + y'^j -2x', -2y', 1. Multiply this by the determinant (which only differs by a numerical factor from the EXAMPLES. 27 preceding) whose four rows are such as 1, x', %fy x*^ + ^, and the first term of the product determinant vanishes, the second being (a/ — a")* + (y* — y")*- ^ *^6^ *^ square of the distance between two of the points be (12)', the product determinant is 0, (I2)«, (13)«, (14)« (21)«, 0, (23j', (24)« (31)2, (32)2, 0, (34)« (41)^ (42)«, (43)' Q =0, which is the relation required. As has been already seen, this determinant expanded gives the weU-known relation (12) (34) + (13) (24) ± (14) (23) = 0. The relation connecting five points on a sphere is the corresponding determinant with five rows. Ex. 12. To find a relation connecting the mutual distances of three points on a line, four points on a plane, or five points in space. We prefix a unit and cyphers to tho two determinants which we multiplied in the last example, thus 1, 0, 0, a;'« + y^, - 2a;', - 2y', 1 &c. 0, 0, 0, 1 1, a?', y', aj^ + y'' dec. = 0, We have now got five rows and only four columns, therefore the product formed, as in Art. 25, witl vanish identically. But this is the determinant 0, 1, 1, 1, 1 1, 0. (12)', (13)', (14)' 1, (21)', 0, (23)', (24)' 1, (31)', (32)', 0, (34)' 1, (41)', (42)', (43)', which is the relation required. If we erase the outside row and column, we have the relation connecting three points on a line ; and if we add another row, 1, (61)', (62)', Ac, and symmetrical column, we get the relation connecting the mutual distances of five points in space. We might proceed to calculate these determinants by subtracting the second column from each of those succeeding, and then the first row from those succeeding, when we get 2 (12)', (12)' + (13)' - (23)', (12)' + (14)' - (24)' (12)' + (13)' - (23)', 2 (13)', (13)' + (14)' - (34)' (12)' + (14)' - (24)', (13)' + (14)' - (34)', 2 (14)' Now the^determinants might have been obtained directly in this reduced but lesa symmetrical form by taking the origin at the point (1), and forming, as in Art. 26, with the constituents ar'y', x"f/', «fec., the determinant which vanishes identically, = 0. a;" + y", xV + y'f, x'x'" + yy " x'x" + y'y", ar"' + y*", af'x'" + y'y " a^x'" + y'y"', x'x'" + y Y", «"" + y'"' which it will readily be seen is equivalent to that last written. Ex, 13. To find the relation connecting the arcs which join four points on a sphere. Take the origin at the centre of the sphere, and form with the direction-cosines of the radii vectores to each point, cos a', cos /3', cos y' \ cos a", dec, a determinant which vanishes identically, this will be 1, cos ad, cosoc, cos 0(2 cos^, 1, coadc, co&hd oo&ca, cosc6, 1, cosed OOSefo, C0B(26, COSe^, 1 =: 0. 28 MINOR AND RECIPROCAL DETERMINANTS. if we Bnbstitnte for eftch cosinei oos oi, 1 — ^^-y + Ac, and then snppoee r the zadias of the sphere to be infinite, we derive from the determinant of this article the xelatioii of last article connecting four points on a plane. Ex. 14. If <p (\) = , calculate 4> (\) . 4> (- X). The new a - X, h, ff, A, J-X, /, determinant is one of like form with X' instead of X, the first line being A — X', Hf Gf Ac, where F = ffh+f{b-hc), G = hf+g{c-\'a\ JT =^ + A (a + ft), and the expanded determinant equated to cypher gives X' — L\^ + 3/X' — N= 0, where X = a« + ft» + c« + 2 (/« + ^* + A«) , Jf = (fttf -/2)2 + (ca - ^«)« + (a* - A«)« + 2 (a/- ^A)« + 2 (fty - A/)« + 2 (cil -^)«, and iVis the square of the original determinant with X in it = ^ Z, Jf, JV are then all essentiallj positive quantities. In like manner if 4> (X) be formed similarly from any symmetrical determinant, <t> (X) <t> {— X) equated to nothing, gives an equation for X^, whose signs are alternately positive and negative, which therefore, by Des Cartes'^ rule, cannot have a negative root. The above constitutes Sylvester's proof that the roots of the equation <^ (\) = are all reaL It is evident, from what has been just said, that no root can be of the form /? ^(— 1), and in order to see that no root can be of the form o + ^ J(— 1), it is only necessary to write a^ a = a'f b — tL = b% p — a^c'f when the case is reduced to the preceding. LESSON IV. MINOR AND RECIPROCAL DETERMINANTS. 27. We have seen (Art. 16) that the minors of any deter* minant are connected with the corresponding constituents by the relation a,-4, + a,-4, + a^A^^ + &c. = A, and these minors are connected with the other constitaenta by the identieal relations \A^ + b^A^ + b^A^ + &c. = 0, c^4i + o^^ft 4 c,4, + &c. = 0, &c. (SOLUTION OF LINEAR EQUATIONS. 29 For Since the determinant is equal to a^A^ + a^A^ 4 &c., and since A^j w4„ &c. do not contain o,, a^, &c., therefore b^A^-^b^A^ + &c. is what the determinant would become if we were to make in it a, = J^, a^ = Jj, &c. ; but the determinant would then have two columns identical, and would therefore vanish (Art. 14). 28. From the above can be deduced useful identical equations connecting the products of determinants formed with the same constituents. Thus writing down the two identical equations (Art. 3) a {be') + b {ca') + c (ai') = 0, a' ( Jc') + &' (ca') + <5' (a&O = ; multiplying the first by d\ the second by d] and, subtracting, we have {ad') {bo') + {bd') {ca') + {cd') {aV) = 0. Similarly from the three equations a {bc'd:')-b {cd'a'')^-c {da'V')^d {aVd')^^ a' (Jc'rTO - V (c(iV') + d {dolV) - d! (aJ V) = 0, d' ( Jc'd") - V (ctfa'O + c" {da'V) - rf" (ai V) = 0, multiplying these respectively by {flf')i \^'f\ {^f)% and adding, we deduce the identity {wlf") (Jc'efO - ^eT) Ka") + {cej") {doCb") - (de'/'O (aJV^^O ; and so on. 29. We can now briefly write the solution of a system of linear equations 0|» + ^,y + Ci« + &c. = f , o.^ + ij^ + Cj^ + &c. = fly, a^ + Jjy + cji + &c. = f, &c., for, multiply the first by -4„ the second by A^^ &c., and add, the coefficients of y, z^ &c. will vanish identically, while the coefficient of x will be a^^, + aj,^^ + &c., which is the deter- minant formed out of the coefficients on the left-hand side of / 30 MINOR AND RECIPROCAL DETERMINANTS, the equation, which we shall call A. Thus we get Aa? = A^^ 4 A^rj + ^3?+ &c., 30. The reciprocal of a given determinant is the determinant whose constituents are the minors corresponding to each con- stituent of the given one. Thus the reciprocal of {afi^c^^) is where ^„ £,, &c., have the meanings already explained. If we call this reciprocal A', and multiply it by the original deter- minant A, by the rule of Art. 22, we get «,^. + *.^. + «. C^., <^A. + ^^« + c, Cf,, a.4. 4 J.B, + c, (7, «A + JA + c, C;, a.^, + &.5, + c.C,, a^A^ + J,5, + c,(7. But (Art. 27) a, 4, + J,B, + c, C. = A, o,^, + i.B, + c, C, = 0, &c This determinant, therefore, reduces to A, 0, = A'. .0, A, 0, 0, A Hence [a^ iA^A) = {<^M" 5 therefore (^.5,C,) = (aA^J*. And in general, A'A = A" ; therefore A' = A*~\ 31. If we take the second system of equations in Art. 29, and solve these back again for ^, 17, &c., in terms of Ao;, Ay, &c., we get A'l = a, Aa; + b^Ay + c^A^ + &c., where a^, bj, c, are the minors of the reciprocal determinant. But these values for ^, 17, ^, &c., must be identical with the ex-> pressions originally given; hence, remembering that A' = A"^, we get, by comparison of coefficients, a, = A"^a,, b, = A'^i,, c, = A"-^c„ &c., which express, in terms of the original coefficients, the first minors of the reciprocal determinant. MINORS AND COMPLEMENTAL MINORS. 31 32. We have seen that, considering anj one column a of a determinant, every element contains as a factor a constituent from that column, and therefore the determinant can be written in the form ^a^p. In like manner, considering any two columns a, b of the determinant, It can be written In the form 2 [ajbg) Ap 9, where the sum 2 {(iphq) Is Intended to express all possible determinants which can be formed by taking two rows of the given two columns. For every element of the determinant contains as factors a constituent from the column a, and another from the column h ; and any term ajbgCrda^ &c., must, by the rule of signs, be accom- panied by another, — a^ipCrd^ &c. Hence we see that the form of the determinant Is 2 (^p^^) Ap g ; and, by the same reasoning as in Art. 16, we see that the multiplier Ap g is the minor formed by omitting the two rows and columns in which Op, bg occur. In like manner, considering any p columns of the determi- nant, It can be expressed as the sum of all possible determinants that can be formed by taking any p rows of the selected columns, and multiplying the minor formed with them, by the complemental minor ; that Is to say, the minor formed by erasing these rows and columns. For example. The sign of each term In the above is determined without diffi- culty by the rule of signs (Art. 8). It Is evident, as in Art. 27, that if we write in the above a c for every J, the sura 2 (a^cj [c^d^e^ must vanish Identically^ since It Is what the determinant would become if the c column were equal to the b column. 33. The theorem of Art. 31 may be extended as follows: Any minor of the order p which can he formed out of the inverse constituents A^^ B^^ &c. is equal to the complementary of the cor" responding minor of the original determinant^ multiplied by the [p — 1)'** power of that determinant. 32 MINOR AND RECIPROCAL DETERMINANTS- For example, in the case where the original determinant is of the fifth order, (J.5.) = A {cM\ iA^A) = A' K«.), &c. . The method in which the general theorem is proved will be sufficiently understood from the proof of this example* We have Ax = A^^ + A^rj + A^^+ A^m + J^v, Therefore A^.* - I^A^ = M.5.) ? + M,5,) ?+ ( ^B.) « + (^.5.) V. But we can get another expression for x in terms of the same five quantities y^ f , ^, co, v. For, consider the original equations, and eliminate z^ Wj Uj we get {a^c^d^e^)x+ [\c^d^e^y^ {c^d^e^) f- (c^^i) ?+ M^) «^- MaO ''J and since [afi^d^e^ is by definition = -B„ comparing these equa- tions with the former, we find [A^B^ = A [c^d^e^^ &c., Q.E.D. Ex. 1. If a determinant yanisb, its minors Ay^ A^ &q , are respectively proportional to Bij B21 Ac. For we have just proved that A^B^ — A^B^ = AC, where C is the second minor obtained by suppressing the first two rows and columns. Jl tbxsBL ^ = 0, we have A^ : A^ : \ B^ \ B^t &o, Ex. 2. A particular example of the above, which is of frequent oocunenoei is obtained by applying these principles to the determinant considered, Ex. 16, p. 17. We thus find, using the notation of that example (j rVj - (o) =^ ^ (" «) » C"** Sunfacn, Art. 80). Ex. 8. As in Art. 32, the determinant {tiyh^^d^ may be written (aifta) M4) + (fli^a) {pM + («i*4) Ma) + W4) ipyd^) + K^s) (<?i<W + W$) (Ci<y . Ex. 4. If six arbitrary quantities p^ q, r, 8,tfU he assumed, and we denote by {ab) the quantity # EXAMPLES. 33 idmilarlj letting lcd)=p {e^ + q {e^) -f r («,d^ + s (Cid^ 4 t (e^^) + u {e^^), Ac^ with like meanings for six such fimctions, then it is eaaij seen by the ident i cal yanishing of groaps, as in Art. 82, when the form in Ex. 8 is written with two or more suffixes the same, that {be) (orf) + («i) (W ) + (a*) (erf ) = (iw + ^« + ni) (fli Vi^J. See Surfaces f Art. 57e. Ex. 5. The homographio relation between two sets of four quantities a, fi, y, i and a'f P'f y'f S* is fomid from the relation of a one-one correspondence between them, viz. Aaa' + Ba + Ca' + i> = 0, 4c. (CMm, Art. 881) in the form of the vanishing of the determinant €ta'f a, a'f 1 Wf 7i y't 1 w, a, a*, 1 Oalcnlating this by Ex. 8, it is found = 08'-y')(a'-aO(^y + aa) + (y'-a')03'-aO(ya+^) + (a'-«(y'-aO(a^+ya), but this is evidently /9y+oa, 1, ffy'+a'V ya + /3a, 1, yV + /3'a' a/9+ya, 1, a'/r + y'd* and Since 03* - yO (a' - a^ + (y' - a') (/S* - aO + (a' - /T) (y' - 30 = 0, the determinant may also be written with the following abbreviations 4=(/3-y)(a-a), B=(y-a)08-a), C = (a -/9) (y - a), ^'=03'-y0(«'-ao, ^' =(/-«') 03' -ao, C"=(a'-/r)(y'-ao, in any of the forms B*C-BC' ^C'A-CA* zzA'B- AB'. This leads to the expansion of the involution determinant (Ex. 7, p. 25) when ^^a'f i' = af in the form there given among others. Ex.6. = 2sinesin(<r + 0) ooeV|, 008 0*1, 1 COS^a,, COS^ty 1 cobV„ COS^s, 1 sin a sin (a + 6), sin a, sin (a + 0), 1 mnfi Bin(/3 + 6), sin/3, sin(/3 +0), 1 sin y sin (y + 0), sin y, sin (y + 6), 1 sin a sin (a +6), sin a, sin (a + 0), 1 = 16 sin e Bin («r + 0) Bin J (/3- y) Bm|(y - o) sini (a - i3) sin J (a - a) sin J(/3 - a) 8ini(y - a), as is easily seen by applying Ex. 3, coupling first and fourth columns and second and third, and introducing the abbreviations 2o' = a + /3 + y + a, 20-1 = a + /3-y-a, 2<r, = a-/3 + y-a, 2a', = a-/3-y + a, Ex.7, The theorem of Art. 83 may be established otherwise as follows. Let =r A, then 6l« ^«f •••^K '!> M» f» ••• ^ L\t Xfj, •••Zfn = A*-* is by Art. 80 the determinant of the reciprocal, or, as Cauchy called it, the adjoint system. P 34 SYMMETRICAL AND SKEW STlilMETRICAL DETEBMINANTS. Now we have the following forms for the product C^) •••V/|| A = 0, B^* B^ ...5ii «r-Bi, -5i» = ajA-'; whence a factor A can be removed. Similarly the product Z/3, ...Z/n I A« = 0, A, 5j, B^f,,.Bm Oj Uj C3J C4J •*.C|| 0, 0, //J, Z/^, •••Z/i» aiBy^a^B^ biBi+b^^t ^t* ^n OiCi+ a^C^ i from which the common factor /&* can be removed. And in like manner generally for a minor of order p. LESSON V. SYMMETRICAL AND SKEW SYMMETRICAL DETERMINANTS. 34. In this lesson it is convenient to employ the double suffix notation, and to write the constituents a„, a,,, &c ; and we, therefore, begin by expressing in this notation some of the results already obtained. We denote the constituents of the reciprocal determinant by a,„ a,,, &c., where, if art be any con- stituent of the original, On is the minor obtained by erasing the row and column which contain that constituent. The equations of Art. 27 may then be written O'rfln + «„«,, + (^n^n + &c. = A, ^ria/i + Orja/^ + «rS«r'« + &C. = 0, or more briefly 2,e7„a„ = A, S,a„a^, = 0; that is to say, the suni of the products «„«/, (where we give every value to 8 from 1 to n) is =0, when r and / are different, and = A when r = »^. Since any constituent a^ enters into the determinant only in the first degree, it is obvious that the factor a„, which multiplies it, is the differential coefficient of the determinant taken with respect to a„; similarly, that the second minor (Art. 32), which multiplies the product of two constituents a^^^ fl„, is the second differential coefficient of the determinant taken with respect to fhese two constituents, &c. «, ©, w U, V, to «1» «1, Wl — ill, r„ w, «2, t^a, Wj «1, «'$» ^8 SYMMETRICAL DETERMINANTS. 35 If any of the constituents be functions of any variable x^ the entire differential of the determinant, with regard to that variable, is evidently a,, -j^ + a^^ ~J^'^ ^^') *°^ ^^7 ^ written down as the sum of n determinants in each of which one row (or column) only, differs from that in the original determinant by having the constituents of the latter differentiated. Ex. If «,, v^ Ac denote the first differentials of u, v, dec with respect to « ; 14, «, the -second differentials, dec, prove dx The differential is the som of the nine prodacts of the differential of each term by the minor obtained by suppressing that term, But the first three terms denote the result of changing in the given determinant the first row into Uy, v,, t^i, and therefore vanish; the second three terms vanish aa denoting the result of changing the second row into «2, tx,, Wj,; and there only remain the last three terms which denote the result of changing the last row into «^ 93, w^. The same proof evidently applies to the similar determinant of the »<& order formed with n functions. S5. The determinant is said to be symmetrical (Art. 10) when every two conjugate constituents are equal [a„^a^. In this case it is to be observed, that the corresponding minors will also be equal (oe„=:a^); for it easily appears that the deter- minant got by suppressing the r*^ row and the 5** column differs only by an interchange of rows for columns from that got by suppressing the s*^ row and the r^ column. It appears from the last article, that if any constituent a^ were given as any fiinction of its conjugate a„, the differential coefficient of the determinant, with regard to a„, would be a„ + a„ -j-^ . In the present case then, where a^^a^ a„ = a^, the differential coefficient of the determinant, with regard to a„, is 2oe„. The differential coefficient, however, with respect to one of the con- stituents in the leading diagonal a„ remains as before o^, since such a term has no conjugate distinct from itself. 36. If, as before, a„ denote the first minor of any deter- minant answering to any constituent a^«, and if /S^ denote the first minor of the determinant o^ answering to any constituent 36 STMHETRICAL A.VD SKEW STHMETSIOAL DETEEMISATtTa. 00, wticli will, of course, be a secoud minor of the origioitl determinaot, then this last may be written where we are to give i every value esccpt r, and k every value except a. For, any element of the determinant which does not contain the constitueat a„ must contain some other constituent from the r" row, and some other from the s" column ; that is to Bay, must contain a product such as a,^^ where i and k are two numbers different from r and a respectively. But as we have already seen, the aggregate of all the terms which multiply (^ is K„; and the coefficient of a^a^ (by Art, 32) differs only ID sign from that of a„«a; that is to say, differs only in sign from the coefficient of a^ in a„. Therefore —ffa is the value of the coefficient in question. Thus then if we have calculated a Bymmetric determinant of the n— 1"* order, we can see what additional terms occur in the determinant of the n"* order. Let A be the determinant, D that obtained by suppressing the outside row and column, /9„ any mbor of the latter, and we have A = Da,. - S,a' j3„ - 2 2„a„«^^ where r is supposed to be different from s, and every value is to be given to r and a from 1 to n — I . Again, we have occasion often, as at p. 17, to deal with determiuauts such as "iD "lit "^m ^j , «„, a„, \ ««, «», \ \, \, X, , obtained by bordering a symmetric determinant horizontally and vertically with the same constituents. This is in fact a symmetric determinant of the order one higher, the last term vanishing, and is - I^jA" + <»nV + i^V + ^<^aW + 2a„>-,\ + 2a„X,\,), or generally - 2^a„V - 2S^„X,\. 37. If any symmetric determinant vanishes, the same deter' minant bordered as in the last article isj with sign changed if SKEW STMMETfilCAL DETERMINANTS. 37 need he^ a perfect square^ when considered as a function of \^ \, \, &c. We saw (Art. 33, Ex. I) that when the deter* minant vaniahea oCuOn^a',,, &c., whence it ia evident that ^1^9 &c. must have all the same sign, and we have generally oCr» = ±V(owO« Further, since it waa ahewn in the aame ex- ample that when a determinant vaniahea, the conatituents of the reciprocal determinant in the aecond row are proportional to those in the firat, it foUowa that the aigna to be given to the radicala are not all arbitrary. If, for inatance, in the above we write «,, = + V(aii«n)) «» = + VCanOi ^^^ ^^ ^^^ forced to give the poaitive aign alao to the aquare root in a„ s V(a„a,,). Substituting, then, theae valuea in the reault of the laat article, it becomea, if o^^, &c. be poaitive, the negative square, - {\ V(au) + \ VCO + \ V(0 + &c.}«, and if a^^, &c. be negative the determinant ia a poaitive aquare. What haa been juat proved may be atated a little differently. We may conaider the bordered determinant aa the original de- terminant ; of which, that obtained by auppresaing the row and column containing X ia a first minor, and a,, obtained by aup- preaaing the next outaide row and column is a second minor. And what we have proved with respect to any symmetrical determinant wanting the last term a^^, is, that if the first minor obtained by erasing the outside row and column vanish, then the determinant itself and the second minor, similarly obtained, must have opposite signs. And this will be equally true if a^ does not vanish. For in the expansion of the determinant, a^ ia multiplied by the first minor, which vanishes by hypothesis, and therefore the presence or absence of a^ does not idSect the truth of the result. 38. A shew symmetric determinant is one in which each constituent is equal to its conjugate with its sign changed. Constituents a„ in the leading diagonal, being each its own conjugate, must in this case vanish ; otherwise each could not be equal to itself with sign changed. A skew symmetrical determinant of odd degree vanishes. For if we multiply each row by — 1 ; in other words, if we change 38 SYxMMETRICAL AND SKEW SYMMETRICAL DETERMINANTS. the Sign of every term, it Is easy to see that we get the same result as if we were to read the columns of the original determinant as rows, and vice versa. Thus, then, a skew symmetrical determinant is not altered when multiplied by (- I)"; and, therefore, when n is odd, such a determinant must vanish. It is easy to see that the minor a„ differs by the sign of each constituent from the minor a„, and therefore a^=(— l)'*'^a^ Hence a^ = a„ when n is odd, and is equal with contrary sign when n is even. a„ is itself a skew symmetric deter- minant, and therefore vanishes when the original determinant is of even degree. The differential coefficient of the determinant, with regard to any constituent a„, being a„ + ol„ -—^ Is a^ - a^. When therefore n Is even it is = 2a„ and when n Is odd It vanishes. 39. Every skew symmetrical determinant of even degree is a perfect square. (Prof. Cayley). It was shewn (Art. 36) that any determinant is and in the present case a„^ vanishes, as does also a^^, which is a skew symmetric of odd degree. On this account therefore we have, as In Art. 37, ^'„ = i8„^„; and therefore, exactly as in that article, the determinant is shown to be Kn ^/{0n) + «^ V(^J + aan V(^J + &0]\ The determinant is therefore a perfect square if ^j„ 13^ are perfect squares, but these are skew symmetries of the order w-2. Hence the theorem of this article is true for deter- minants of order n If true for those of order n — 2, and so on. But it is evidently true for the determinant of the second order 0, a 12 , which is = a'j,. Hence it Is generally true. 40. We have seen that the square root of the determinant contains one term «^.,,« V(/8„.,^„_J, where /3^_i,„.i contains no constituent with either of the suffixes n - 1 or n. But taking any two of the remaining suffixes, such as n - 3, n - 2, we see that SKEW SYMMETRICAL DETERMINANTS. 39 \/(y9„-,,n-t) contains a term ^n-s,-. V(7„.8. -J) ^^erc 7,.,,,^ con- tains none of the four suffixes, of which account has already been taken. Proceeding in this manner we see that the square root will be the sum of a number of terms such as a,<a^^a^...a-, ; each of which is the product of ^n constituents, and in which no suffix is repeated. The form, however, obtained in the last article «,„ V(^,i) ± ««» V(^m) + &c. does not show what sign is to" be affixed to each term. Thus if the method of the last aiticle be applied to the skew symmetric of the fourth order, its square root appears to be «,8«84 ± ^i3^«4 ± ^m^m J ^^* ^* has not been shown which signs we are to choose. This, however, will appear from the following considerations : When in the given determinant we interchange any two suffixes 1, 2, since this amounts to a transposition of the first and second row, and also of the first and second column, the determinant is not altered. Its square root then must be a function, such that if we interchange any two suffixes it will remain unaltered, or at most change sign. But that it mil change sign is evident on considering any term a,,aj^, &c. which, if we interchange the suffixes 1 and 2, becomes a„a^j &c. ; that is to say, changes sign, since a,, = — a,,. It follows then, in the particular example just considered, that the signs of the terms are «,8«84 ^ ^i8^«4 + ^u^ss 5 ^^^ *^ ^c give the second term a positive sign, the interchange of 2 and 3, which alters the sign of the last term, would leave the first two unchanged. And generally the rule is, that the square root is the sum of all possible terms derived from a,,«3,...a„.,^„ by interchange of the suffixes 2, 3, ...n, where, as in determinants, we change sign with every permutation. But it is possible, and the better course is, to effect the interchanges in such manner that the signs shall be each of them + ; thus, in the particular example, the expression may be written a^^a^ H- a„«4a + «,4^8a- Ex. 1. To write out the skew Bymmetrical determinant of the sixth order. Denoting by (1234) the expression just given with all signs positire for the square root of that of the fourth order, if we proceed always in the same cyclical order we write down the expressions (3456), (4662), (6623), (6234), (2346), and the expression required is the square of the sum of the fifteen terms contained in «»« (8466) + a„ (4662) -ha^ (6623) + 0,4 (6234) + a^ (2346). 40 SYMMETRICAL AND SKEW SYMMETRICAL DETERMINANTS. Ex. 2. If the constitaents of a symmetrical detenmnant be each increased bj k times the corresponding constituents of a skew symmetrical, the determinant so formed when expanded contains only eyen powers of Jt, For, altering k to ~ k does not change its yalue. For instance (see SurfaceSf Art. 80c) we may write a, h^kZf g + ky, l — ka h + hsj 6, f— kXf m— k^ SI — kt/j f+ kxy Of n — ky I +A:a, «H- A"/3, n + ky, d. The same proof shews that if conjugate constitaents of a determinant Imfunagmaiy quantities, the yalue of the determinant is real. Ex. 8. The square of any determinant of eyen order may be written as a skew symmetrical determinant. For instance, calculate the product a„ Oj, o,, a^ Jj, b. Cif e. dif d^f rf|, d^ 21 *» ^4 «4> -«»««>- «1 ^4» "~ ^» <'2» ~" <^l and it is found to be of the skew symmetrical form. Or, again, for any multiplien Pf 9t ^1 *f ^t ^f ^orm the determinant product of and multiply it by 0, -r, q,-i r, 0,-py-t «, «l» ««» «•» »4 *1» ^2» *» ^4 '!» »'2» ^»» ^4 C?l, dj, rf„ d^ <*!> ^> «8> «4 K *2i *»» '!> ''2> *'«» ^4 0* the product may be written ^»> ^ <^» <^4 , (oi), (a<j), (arf) (5a), , {b€\ {bd) {ca\ {eb), , icd) (da), (d&), (cfc), in the notation of Ex. 4, Art. 33. Since {off) = — {ba) Ac, we haye thus proyed {(be) (ad) + (ca) {bd) + {ab) (ed)}^ ={pi + ^ + ruy (aJf^M^ which was otherwise seen in that Example. 41. We can reduce to the calculation of skew symmetric determinants the calculation of what Prof. Cayley calls a skew determinant, in which, though the conjugate terms are equal with opposite signs, a^ = — a^,., yet the leading terms a,,., a^^, &c. do not vanish. We shall suppose, for simplicity, that these leading terms all have a common value X. We prefix the following lemma : If in any determinant we denote hy D the result of making all the leading terms =0, by Di what the minor corresponding to an becomes when the leading terms are all made = 0, by D^^ what the second minor corresponding to ^ifikk becomes when the leading terms vanish, &c.| then the JSKEW DETERMINANTS. 41 giveD determinant, expanded as far as the leading constituents at6 concerned, is A = i> + Sa«J9, + ^iflu^a + . . .+ a^,d^. . .a,^, where, In the first sum t has any value from 1 to 7t, in the second sum ty h are any binaiy combinations of these numbers, &c^ For, the part of the determinant which contains tlo leading constituent is evidently D ; the terms which contain an are auAu^ where A^ is the corresponding minor, hence the terms which contain a^ and no other leading constituents are got by making the leading constituents = in A^ ; and so for the other terms. 42. If this lemma be applied to the case of the skew deter* minant defined in the last ai'tide, all the terms i>,-, D^^ &c. are skew symmetric determinants ; of which, those of odd order vanish, while those of even order are perfect squares. The term a,fl^...a^^ is X", and the determinant is A = X* + X"""SZ>, + X"-*SJ5, + &c., where i>,, i>^, &c. denote skew symmetrical determinants of the second, fourth, &c. orders formed from the original in the manner explained in the last article. Ex. 1. I where 0^^ = — Oi^ dEO> is = \» + X (ai,« + a„« + 0^*). Ex. 2. The similar skew determinant of the fourth order expanded is X« + \« (aj2« + o„« + a,4« + a„* + 084* + o^*) + {a^^h* + ^it^m + ai4«»)*. 43. Prof. Cayley {CreUcj vol. xxxiL, p. 119) has applied the theory of skew determinants to that of orthogonal substitu- tions, of which we shall here give some account. It is known (see Surfaces^ p. 10) that when we transform from one set of three rectangular axes to another, if a, 5, c, &c. be the direction-cosines of the new axes, and if that we have X* + F* H- Z" = a;* + / + z\ whence a' + a'* + a"* = l, &c., aft + a'6' + a"6" = 0, &c. ; that also we have x^aX^-a^Y-Va^'Z, y^hX\VY^V'Z, z^cX^c'Y-^c''Z, 42 SYMMETRICAL AND SKEW STMMETBICAL DETERMINANTI9. and that we have the detenninaiit formed by a, J, c; a', 6', c'; &c. = ±l. It Is also useful (In studying the theory of rotation for example) instead of using nine quantities a, h^ c, &c. connected by six relations, to express all in terms of three independent variables. Now all this may be generalized as follows: If we have a function of any number of variableS| it can be transformed by a linear substitution by writing and the substitution Is called orthogonal If we have a* + »* + «" + &c = X« + r* + Z« + iSbc., which Implies the equations ^11' + ^«* + &^* = h ^ii^u + ««i^M + &c. = 0, &c. Thus the n* quantities a,^, &c. are connected by ^n (n + 1) re- lations, and there are only ^n (n — 1} of them independent.] We have then conversely equations which are immediately verified by substituting on the right-hand side of the equations for a?, y, «, &c. their values. And hence, the equation X' + y* + &c. = a5'+y* + &c. gives us the new system of relations «ii' + «n* + &c. = 1 , a,,a,j + a^^a^ + &c. = 0. Lastly, forming by the ordinary role for multiplication of determinants, the square of the determinant formed with the n' quantities a„, &c., each constituent of the square vanishes except the leading constituents, which are each = 1. The value of the square Is therefore = 1. Thus the theorems which we know to be true in the case of determinants of the third order are gene- rally true, and it only remains to shew how to express the n' quantities in terms of Jw (n - 1) independent quantities. This we shall effect by a method employed by M. Hermite for the more general problem of the transformation of a quadric function into itself. See his paper *Kemarques, &c.,' Qmb. and Dub. Math Jour.^ vol. ix. (1854), p. 63. ORTHOOONAL SUBSTITUTIONS. 43 44. Let us suppose that we have a skew determinant of the (n - 1)**^ order, J„, J^,, &c. where h^-- i^ a°d J„ = J„ = i^ = 1 J and let us suppose that we form with these constituents the two different sets of linear substitutions, from adding which equations we havCi in virtue of the given relations between J^j, J,,, &c., If now the first set of equations be solved for f , i;, &c. in terms otxj/j &c., we find, by Art. 29, Af = fi^^x + /9^ + fi^^z + &c., Ai; = fi^^x + fi^ + &c. (where )8,„ )8,,, &c. are minors of the determinant in question) ; and putting for 2f , x + X^ &c., these equations give AX= {2/3,, - A) a; + 2/3„y + 2/3„« + &c., A r= 2/3,,» + (2)8„ - A) y + 2)8,,« +&c., &c., which express X, F, &c. in terms of a?, y, &c. But if we had solved from the second set of equations for ^, 17, &c. in terms of X and F, &o. we should have found AS^l3^^X+l3^^Y+/3^Z+&c., Ai; = )8.,X+/S„r+)8^+&c., whencCi as before, A» = (2/3„ - A) X+ 2/3„r+ 2/3^^+ &c., Ay = 2/3^X + (2/S„ - A) Y-¥ 2/3„Z+ &c. Thus, then, if we write — "^^ -«ii> — ^;^ =««> -^ «w» ^ ^^^w we have a?, y, &c. connected with X, F, &c. by the relations X = a^jX + a^Y-\- &c., y = a^^X + a„ F+ &c., &c., X=aj,aj +«„y +&C, Y=a^^x + a^ +&c., &c. We have then a?, y, &c., X, F, &c. connected by an orthogonal enbstitution, for if we substitute in the value of x the valuer 44 SYMMETRICAL AND SKEW SYMMETRICAL DETERMINANTS. of JT, Yj &c. given bj the second set of equations, in order that our results may be consistent, we must have «ji* + ««* + «ni* + &©• = 1 , a^fi^^ + a^^a^ + a^fi^ + &c. = 0, &c. Thus then we have seen that taking arbitrarily the ^n (n - 1) quantities, b,,, 5,,, &c., we are able to express in terms of these the coefficients of a general orthogonal transformation of the n** order. « Ex. X, To fonn an orthogonal transformation of the second order. Write A= 1, X -X, 1 =1 + X«, then /9ii r= /3„ = 1, j3^ = X, /^tt = — ^» and our transformation is (1 + X«)aj= (I-X2)Z+2Xr, (l + X2)Z=(l-X2)aj-2Xy, (l + X«)y = -2XJr+(l-X«)r, (1 + A2)r= 2X0? +(1-X«)y. Ex. 2. To form an orthogonal transformation of the third order. Write A= 1, Vt "11 -V, 1, X Ml "K I Then the constitnents of the reciprocal system are I + X', p + \fif — /* + Xv — V + \fif 1 + ft', X + fiw /* + Xy, - X + /ttw, 1 + w« consequently the coefficients of the orthogonal sabstitntion hence deriyed are l + \«-^t«^7^ 2(i( + X/*), 2(Xv-./.*), 2(X/ii-y), l + /i»a-X«-v», 20ii' + X), 2{\ii + /.i), 2{fiv\), i + v2-X«-/i»«, where each term is to be divided l^ 1 + X' + /u' + v^* 45. It is easy to see that for a symmetrical determinaat of fhe orders 1, 2, 3, 4 the member of distinct terms is =1, 2, 5, 17 respectively, and the question thus arises what is the number of distinct terms in a symmetrical determinant of the order n. This number has been calculated as follows by Professor Cayley : = 1 + X« + fi« + p*. * The geometiic meaning of these coefficients may be stated as follows : Write X = a tan j|6, fi = b tan j^O, v=zc tan ^6, then the new axes may be derived from the old by rotating the system through an angle romid an axis whose direction-cosinfis are a, b, c. The theory of orthogonal enbstitntions was fot investigated by Enler {Nov, Comm, Petrop., vol. xv., p. 76, and vpl. xx., p. 217), who gave fprmnls for the transformation as far as the fourth order. The quantities X, ft, v, in the case of the third order, were introduced by Rodriguee, Lioumlle, voL v., p. 405. The genenJ theory, explained above, connecting linear transformations with skew determinants, was given by Cayley, CrelU^ voL xxxii., p. 119. NUMBEK OF TEfiMS. 45 Consider a partially Bymmetrlcal determinant represented (see Art. 11) bj the notation aa bb cc pp bb pp' where in general fg^gf^ but all the letters jo, j, ... are distinct from all the letters p'^ /, ... so that these letters give rise to no equalities of conjugate terms ; say, if in the bicolumn there are m rows aa^ bb^ ... and n rows pp\ qc[^ ... this is a determinant (m, n] ; and in the case n=0, a symmetrical determinant. And let ^ (m, n) be the number of distinct terms in a determinant (m, n). Consider j/?r«^ a determinant for which n is not 0, for instance aa,^ abj ap j agf buj bb^ bp\ b^ i>«) p^ pp'^ pi qa^ qb^ gp\ j/ then qa^ qb^ qp\ qg[ are distinct from each other and from every other constituent of the determinant, and the whole determinant is (disregarding signs) the sum of these each multiplied into a minor determinant ; the minors which multiply qa^ qb are each of the form (1, 2) ; those which multiply qj^^ qc[ are each of the form (2, 1) ; and we thus obtain ^(2, 2) =2^ (1,2) + 2^ (2,1), and so in general ^ (m, n) = m^ (w — 1, n) + n^ (?n, n — 1), whence in particular ^ (7/1, 1) s= m^ (w — 1) 1) + ^ ^1 0). Next, if n = 0, let us take for instance the symmetrical de- te^ii^ant oa J ^ aa^ ab^ ac^ ad bb Ja, bbj bCj bd cc dd ca^ cbj ccj cd da^ dbj dc^ dd 46 SYMMETRICAL AND SKEW SYMMETRICAL DETERMINANTS. We have here terms multiplied by dd; ad,da^ hd.dbj cd.dc; and bj the pairs of equal terms ad.db + bd.dajad.dc-^cd.da^ bd.dc + cd.db^ the other factors being in the three cases minors of the forms (3, 0), (2, 0), and (1,1) respectively ; thus we have <l> (4, 0) = (3, 0) + 3^ (2, 0) + 3^ (1, 1), and so in general 4> (wi, 0) = ^ (« - 1, 0) + (m- 1) (m-2, 0) + 4(w-l)(m-2)^(m-3, 1), which last equation combined with the foregoing <l>{m^ l) = w^(wi — 1, l) + ^(wi, 0) gives the means of calculating (m, 0), ^ (m, 1) ; and then the general equation ^(m, «) = fw^(fn— 1, w) + n^(mi n— 1) gives the remaining quantities 4> (^) ^)* It is easy to derive the equation 2#(aw, 0)-^ (m-1, 0)-(7»-l) ^(w-2, 0)= ^ (wi-1, 0) 4- (w-l)^(7n-2,0) +(7?i-l)(w-2) 4> (7n-3, 0) + + (m-l)...3.2.1^(0,0). And hence, using the method of generating functions, and assuming « = * (0, 0) + f * (1, 0) + ^2 * (2, 0)...+ -j-|^ ^ (m, 0) +..., du u ax 1 — a; that is 2— = fl+a; + -^)dir, u \ 1 - 0?/ ' or, integrating and determining the constant, so that for x^O u shall become s 1, we have u = we find at once 2-^ — m — a?w = ~ , V(l-aj)' whence ^ (tti, 0), the number of terms in a symmetrical deter- minant of the order m, is s 1 .2...m coefficient of oT in V(i - i») • NUMB£K OF TEBMS. 47 The numerical calcnlation bj this formula is, however, somewhat complicated; and it is practically easier to use the equations of differences directly. We thus obtain not only the values of ^ (m, 0), but the series of values * (0, 0), ^ (1, 0), if, (0, I), ^(2,0), ^(1,1), ^(0,2), 4> (3, 0), &C., vbich are found to be h h 2, 2, 2, 5, 6, 6, 6, 17, 23, 24, 24, 24, 73, 109, 118, 120, 120, 120, 388, 618, 690, 714, 720, 720, 720, &c., &c., as is easily verified. But as regards i>{m^ 0) we nray, by writing the equation in u under the slightly difTerent form obtain from it the new equation of differences ^(9n, 0) = wi^(m-l, 0)-i(w-l)(w-2)^(w-3, 0), and the process of calculation is then very easy. Starting from the values ^ (1, 0) = 1, ^ (2, 0) =2, which imply ^ (0, 0) = I, we have w = l, 1 = 1. 1 = 2, 2 = 2. 1 = 3, 5=*3. 2- 1.1, = 4, 17=4. 5 - 3.1, = 5, 73 = 5.17- 6.2, = 6, 388 = 6.73-10.5, &c., &c. ( 48 ) LESSON VI. DISOBIMmATINQ STMMETBICAL DflTEBMlNANtd. 46. If we add the quantity X to each of the leading terms of a symmetrical determinant, and equate the result to 0| we have an equation of considerable importance in analysis.* We have already given one proof (Sylvester's) that the roots of this equation are all real (Ex. 14, p. 28), and we purpose in this Lesson to give another proof by Borchardt (see LiouvUle^ vol. XII. p. 50), chiefly because the principles involved in this proof are worth knowing for their own sake. First, however, we may remark that a simple proof may be obtained by the application of a principle proved in Art. 37. Take the de- terminant a^,+X, a,„ a,8, &c. &c. and form from it a minor, as in Art. 37, by erasing the outside line and column ; form from this again another minor by the same rule, and so on. We thus have a series of functions of \, whose degrees regularly diminish from the n** to the 1**; and we may take any positive constant to complete the series. Now, if we substitute successively in this series any two values of X, and count in each case the variations of sign, as in Sturm's theorem, it is easy to see that the difference in the number of variations cannot exceed the number of roots of the equation of the n** degree which lie between the two assumed values of X, This appears at once from what was proved in Art. 37, * It oocors in the determination of the secular inequalities of the planets (see Laplaoe, Micaniqw CelesCe, Part i., Book ii., Art. 56). Sturm's functions. 49 that if \ be taken so as to make any of these miDors vanish , the two adjacent functions in the series will have opposite signs. It follows, then, precisely as in the proof of Sturm's theorem| that if we diminish X regularly from + oc to — oo , when X passes through a root of any of these minors, the number of variations in the series will not be affected ; and that a change in the number of variations can only take place when X passes through a root of the first equation, namely, that in which X enters in the 'riP^ degree. The total number of variations, there- fore, cannot exceed the number of real roots of this equation. But, obviously, in all these functions the sign of the highest power of X is positive ; hence, when we substitute + oo , we get no variation ; when we substitute — oo , the terms become alternately positive and negative, and we get n variations; the equation we are discussing must, therefore, have n real roots. It is easy to see, in like manner, that the roots of each function of the series are all real, and that the roots of each are interposed as limits between the roots of the function next above it in the series. 47. It will be perceived that in the preceding Article we have substituted, for the functions of Sturm's theorem, another series of functions possessing the same fundamental property, viz. that when one vanishes, the two adjacent to it have opposite signs. Borchardt's proof, however, which we now proceed to give, depends on a direct application of Sturm's theorem. The first principle which it will be necessary to use is a theorem given by Sylvester [Philosophical Magazine^ December, 1839), that the several functions in Sturm's series, expressed in terms of the roots of the given equation, differ only by positive square multipliers from the following. The first two (namely, the function itself and the first derived function) are, of course, (a; — a) (a: — /8) [x — 7) &c., S (a; - ^) (a; — 7) &c. ; and the remaining ones are S(a-)8)'^(aj-7)(aj-8)&c. ; 2(a-)8)''(/3-7)»(7-a)'(a;-8)&c.,&c., where we take the product of any A factors of the given H 60 DISCRIMINATING SYMMETRICAL DETERMINANTS. equation, and multiplying by the product of the sqt(ai*ed of the differences of all the roots not contained in these factors, form the corresponding symmetric function. We commence by proving this theorem.* 48. In the first place, let U be the function, V its first derived function, JS,, JS,, &c., the series of Sturm's remainders ; then it is easy to see that any one of them can be expressed in the form AV-BU. For, from the fundamental equations U=QJ-B,, V=Q,B,-B,, E,= Q,B,- B,, &c, we haTo B^^Q,V^U, and so on. We have then in generalf B^^AV—BUj where, since all the Q^a are of the first degree in Xj it is easy to see that A is of the degree A — 1, and B of the degree A— 2, while J?^ is of the degree n — k. But this property would suffice to determine JB^, jB,, &c., directly. Thus, if in the equation E^ = Q^ F— ?7, we asstime ^, = aaj + J, where a and b are unknown constants, the condition that the coefficients of the highest two powers of x on the right- hand side of the equation must vanish (since B^ is only of the degree n - 2) is sufficient to determine a and b. And so in • I suppose that Sylvester must have originally divined the form of these functions from the characteristic property of Sturm's functions, vl(s. that if the equation has two equal roots a = /3, every one of them must become divisible by x^ a. Consequently, if we express any one of these functions as the sam of a number of products {x — a) (x — /3) <fec., every product which does not indlude either X — a or x — fi must be divisible by (a ~ /3)* j and it is evident in this way that the theorem ought to be true. The method of verification here employed does not differ essentially from Sturm's proof, LiouvillCj vol. vii. p. 356. t The theory of continued fractions, which we are virtually applying here, shews that if we have Iik = AkV— BkU, Rk+i = -4i+iF— -Bt+jC^, then AtBt+i — Au^iBk k constant and =1. In fact, since Rje+i = QkRk — Rk-u we have -4*+! = QkAi — Ak-ii Bk+i = QkBt — Bi^i, whence AkBki i — Ak^Bk — Ak-iBk — AkBk-i, and by taking the values in the first two equations above, namely, where ife = 2 and X; = 8, we see that the constant value = 1. Sylvester's forms. 51 general, if in the function AV—BU we write for A the most general function of the [k - 1)*** degree containing k constants, and for B the most general function of the {k — 2)** degree con- taining A; — 1 constants, we appear to have in all 2^-1 constants at our disposal, and have in reality one less, since one of the coefficients may by division be made =1.* We have then just constants enough to be able to make the first 2^-2 terms of the equation vanish, or to reduce it from the degree n + A; — 2 to the degree n — A. The problem, then, to form a function of the degree n — A, and expressible in the form AV—BU^ where A and B are of the degrees A; - 1, A — 2, is perfectly definite, and admits but of one solution. If, then, we have ascertained that any function B^ is expressible in the form AV— BU^ where A and B are of the right degree, we can infer that B^ must be identical with the corresponding Sturm^s remainder, or at least only differ from it by a constant multiplier. It is in this way that we shall identify with Sturm's remainders the expressions in terms of the roots. Art. 47. 49. Let ns now, to fix the ideas, take any one of theoo functions, suppose S (a - /Sj* (/3 - 7)' (7 ^ ar (a; - S) (x - 8) &c., and we shall prove that it is of the form AV—BUj where -4 is of the second degree, and B of the first in x. Now we can immediately see what we are to assume for the form of ^, by making a? == a on both sides of the equation. The right-hand side of the equation will then become u4 (a - ^) (a - 7) (a - 8) (a - e) &c., since U vanishes ; and the left-hand side will become 2(a-/8)«(^-7r(7-ar(a-8)(fl^-eJ&c. It follows, then, that the supposition a? = a must reduce A to the form S(i8-7)*(a-^)(a-7), and it is at once suggested that we ought to take for A the symmetric function S(/8-7)'(a!-/8)(a!-7). * Just a^ the six constants in the most general equation of a conic are only equiyalent to five independent constants, and only enable us to make the curyQ satisfy five conditions. 52 DISCRIMINATING SYMMETRICAL DETERMINANTS. And in like manner, in the general case, we are to take for A the Bymmetric function of the product of Aj — 1 factors of the original equation multiplied by the product of the squares of the differences of all the roots which enter into these factors. It will not be necessary to our purpose actually to determine the coefficients in By which we shall therefore leave in its most general form, Let us then write down S(a-/3)«(/3-7)«(7-ar(a:^S)&c. = S(a-/3)«(a:-a)(aj-/3) X Si(ar — i8) (a; — 7) &c. + {ax + J) (a; - a) {x - y3) &c., which we are to prove is an identical equation. Now, since an equation of the p** degree can only have p roots, if such an equation is satisfied by more than p values of a;, it must be an identical equation, or one in which the coefficients of the several powers of x separately vanish. But the equation we have written down is satisfied for each of the n values x=o^ x=Py &c., no matter what the values of a and h may be. And if we sub- stitute any other two values of cc, then, by solving for a and b from the equations so obtained, we can determine a and &, so that the equation may be satisfied for these two values. It is, therefore, satisfied for n + 2 values of a?, and since it is only an equation of the (n+1)** degree, it must be an identical equation. And the corresponding equation in general, which is of the degree w + A— 1, is satisfied immediately for any of the n values a? = a, &c. ; while B being of the degree A — 1 we can determine the h constants which occur in its general expression, so that the equation may be satisfied for k other values ; the equation is, therefore, an identical equation. 60. We have now proved that the functions written in Art. 48 being of the form AV—BU are either identical with Sturm's remainders, or only differ from them by constant factors. It remains to find out the value of these factors, which is an essential matter, since it is on the signs of the functions that everything turns. Calling Sturm's remainders, as before, ^„ i?„ &c., let Sylvester's forms (Art. 47) be T,, 7;, &c., then we have proved that the latter are of the form T^ = \fi , T^ = Xgi?^, <&c., and we want to determine \y \, &c. We can Sturm's theorem. 63 at once determine X, by comparing the coefficients of the highest powers of aj on both sides of the identity T^^A^V^B^U] for a;* does not occur in T^, while in V the coefficient of «"'* is n, and the coefficient of x Is also n in A^^ which = 2 (a; — a) ; hence B^ — n*. But the equation T^ = A^V-BJJ must be identical with the equation B^ = Q, F— U multiplied by X, ; we have, therefore, X^ = w*. To determine in general X^ it is to be observed that since any equation T^^A^V—B^U is X^ times the corresponding equation for JS^, and since in the latter case it was proved (note, Art. 48) that A^B^^^-- A^^^B^ = \^ the corresponding quantity for jT^, T^^^ must = X^X^^^. Now from the equations we have A^,T, - A,T^, = (^ A. - ^».A) U^ \K. U. Comparing the coefficients of the highest powers of x on both sides of the equation, and observing that the highest power does not occur in -4^7]^^, we have the product of the leading coefficients of -4^^ and 2\ = X^X^^,. But if we write 2 (a - /8)« = p., S (a - /8)' (a - 7)" (/3 - 7)" =i'., «&c, we have, on inspection of the values in Arts. 47, 49, the leading coefficient in T^=p^^ in ^3 = ^3, &c., and in -4,=sw, in ■^8 =-Ps) ^^ ^A ^-Pa? ^^' Hence Pi'^KK Pb^\K Pi= Wi &C-) whence X3=-Sl^ x^ = ^^ , &c. The important matter, then, is that these coefficients are all positive squares, and therefore, as in using Sturm's theorem we are only concerned with the signs of the functions, we may omit them altogether. 51. When we want to know the total number of imaginary roots of an equation, it is well known that we are only con- cerned with the coefficients of the highest powers of x in Sturm's functions, there being as many pairs of imaginary roots as there are variations in the signs of these leading terms. And since the signs of the leading terms of T,, T3, &c. are the same as those of JS,, £3, &c., it follows that an equation has as 54 SYMMETRICAL DETERMINANTS. many pairs of imaginary roots as there are variations in the series of signs of 1, w, S (a - ^)^ 2 (a - /S)" (/3 - 7)' (7 - a)*, &c. This theorem may be stated in a different form by means of Ex. 3, Art. 26, and we learn that an equation has as many pairs of imaginary roots as there are variations in the signs of the series of determinants h *0» *0» *i J *o) *t) ^ » *0» *1) *S» *8 «1) «2 *1) '^5 *3 *t» *a) ^8) *4 *a» *8> *4 *8l ^8? ^4) ^6 *8J *4) ^6? *6 , &c., the last in the series being the discriminant ; and the condition that the roots of an equation should be all real is simply that every one of these determinants should be positive. 52. We return now, from this digression on Sturm's theorem, to Borchardt's proof, of which we commenced to give an account, Art. 47 ; and it is evident that in order to apply the test just obtained, to prove the reality of the roots of the equation got by expanding the determinant of Art. 46, it will be first necessary to form the sums of the powers of the roots of that equation. For the sake of brevity, we confine our proof to the determinant of the third order, it being understood that precisely the same process applies in general; and, for con- venience, we change the sign of X, which will not affect the question as to the reality of its values. Then it appears im- mediately, on expanding the determinant, that s^ = a„ H" ^m + ^m since the determinant is of the form X' - V (a„ + Og, + ^as)"*" ^^ And in the general case s^ is equal to the sum of the leading constituents. We can calculate s^ as follows : The determinant may be supposed to have been derived by eliminating a?, y, g between the equations Xa:=a,ja?+a„y+a„0, Xy=a,,a;+a^y+a^0, Xz^a^.x+a^y-^a^z. Multiply each of these equations by X, and substitute on the right-hand side for \a?, Xy, \z their values, thus we get >^*«=(«,i*+«,»*+«i8')^+(«n«8i+«,.««ii+«i3«ia^i^+(«^^^^^ borchardt's investigation. 55 from these eliminating a?, y^ z^ we have a determinant of form exactly similar to that which we are discussing, and which may be written Then, of course in like manner, ^» = K + *« + *88 = < + < + < + 2a„« + 2a J + 2a^^. The same process applies in general and enables us from s^ to compute 5^^,. Thus suppose we have got the system of equations "hra^d^^x-Vd^jZ-i-d^^Zy \^y=d^^x-\-d^^rd^z^ ^'«=^8i^+^«iy+^«^> from which we could deduce, as above, 8^ = d^^ "*" ^m + ^ss 5 ^^^^ multiplying both sides by \, and substituting for \x^ &c. their values, we get -hT^x = [d^^a^^ + rf,,a„ + ^,3^,3) x + [d^fl^^ + d^^a^ + £?,3a J y X^^z = (<73,a,, + e73,a„ + d^a^^j x + (rf3,a,, + d^a^ + ^330,,) y + (^81^81 + ^89«88 + ^88^ ^) whence /r^, = rf,,a„ + rf^^a^ + ^33^33 + 2i„a,, + 2^^033 -h 2rf3,a3,. 53. We shall now shew, by the help of these values for 9^, &c , that each of the determinants at the end of Art. 51 can be expressed as the sum of a number of squares, and is therefore essentially positive.* Thus write down the set of constituents 1, 1, 1, 0, 0, 0, 0, 0, ^ll> ^29» ^88? ^28» ^81? ^ia» ^tSJ ^8l) ^12 then it is easv to see that ^0) *1 ^1) «8 is the determinant formed * M. Kummer first found oat by actaal trial that the discriminant of the cubic which determines the axes of a surface of the second degree is resolvable into a sum of squares. {CrelUy yol. XXTI., p. 263). The general theory given here is due, as we have said, to Borchardt. 56 SYMMETRIC FUNCTIONS. from this, by the method of Art. 24, which expresses it as the sum of all possible squares of determinants wliich can be formed by taking any two of the nine columns written above. The determinant «0^ «1 Is thus seen to be resolvable into the sum of the squares K - ^J + K - « J' + K - ««)" + 6 {a J + a3,« 4 a,,*), and is therefore essentially positive. Again, if we write down 1, 1, 1, 0, 0, 0, 0, 0, ^1lJ ^98» ^889 ^831 ^81? ^12l ^SS? ^81? ^18 *11? *88l *88» *88» *8lJ ^8? K^ Ki^ K where h^^^ &c. have the meaning already explained, it will be easily seen from the values we have found that »«. «.. «, «,. «.> ». «.' ««> «4 is the determinant which, in like manner, is equal to the sum of the squares of all possible determinants which can be formed out of the above matrix. And similarly in general. LESSON VII. BTMMETBIC FUNCTIONS. 54. We assume the reader to be acquainted with the theory of the symmetric functions of roots of equations as usually given in works on the Theory of Equations. Thus, we suppose him to be acquainted with Newton's formulce for calculating the sums of the powers of the roots of the equation SYMMETRIC FUNCTIONS. 57 VIZ. «,-;?, = 0, «,-i>A + 2/>, = 0, «8-PA + ft«i-¥. = ^i &C-) whence /r, =/>„ «, =p^ - 2p„ «, =;?,• - 3/>,/^, + 3p„ i&c., and with the formulao 2a*^7» = Vp«, - 5.^5, - 8^^s, - «,,^. + 2j?^.„ &c. We can thus calculate Sa^/S**, &c. first in terms of the sums «„ *,, &C9 and ultimately in terms of the coefficients^,, p^^ &c.* Ex. We can get determinant ezpreflsions for the snms of powers in terms of the coefficients, or vice vend, by solving, as in Art 29, the system of linear eqaationa above written, for «|, «^ Ac, or/?,, p„ Ac. Thus we have h = Pit 1 [ ^P»Pi \i h = Pif 1» 2p» Pi, 1 %>» Pit Pi f •« 8a = ; Ao. 1.2.p,= »if 1 ; 1.2.8.p,= »1, 1, H» »i, 2 h> H, »i Pu 1, 0, 2p» Pm 1, 8Pt» P«» Pi, 1 *P4i />», Pr P\ *i, 1, 0, »» »i, 2, '•» '» '1, ° '«, '» *» 'i , 1.2«8*4.p4 = ; Ac 55. It is more natural to start from the equations * This process is a very bad one, in fact if it were employed thiooghoiit, wt ahonld have for instance to calculate 'La^y, that is p^ from the formula 6^0/37= *,» = p^ - 8«,«, - 8/>, (p,« - 2ig + 2#i +2(Pi»-8/>iP. + 8p.) bnt the process introdnces terms p^^ and pxp^ each of a higher order than />, (reckoning the order of each coefficient as unity) with numerical coefficients which destroy each other. And so again £a*/3 would be calculated from the formula = PiP« - %>•, but there is here also a term p^ of a higher order, with numerical coefficients which destroy each other. And the order in which the several expressions are derived, the one from the other is a non-natural one; «, is required for the determination of £a*/3, whereas it is properly £a*/3 which leads to the value of $^ I 58 SYMMETRIC FUNCTIONS. and thence derive the set3 of equations Px = Sa, A = SaiS, />.' = 2a* + 22a)3, Pz = 2a/37, i'.Ps t 2a*/9- f 32a)87, j^,' =2a'+3Sa'i8+62a/37; we thus have 1 equation to give 2a; 2 equations to give 2a)9 and 2a* ; 3 equations to give 2a^7, Sa'^/S, 2a', and so on. And taking, for instance, the third set of equations, the first equation gives 2aj87, the second then gives 2a^/8, and the third then gives 2a' ; thus 2a«^=p,p,-3;?3, 2a» =p,' -3(p,p,-3;?3)-6^„ =i^i' -¥iP2 +3^3- The process for the formation of the successive sets of equations is further explained and developed in Prof. Cayley's " Memoir on the Symmetric Functions of the Boots of an Equation," PML Trans.j vol. CXLVii, (1857), and the original and inverse sets of equations, for equations up to the order 10, are tbei:eiii exhibited in the form of tables (see Appendix). 56. If we have any homogeneous function of the coefficients i^i» i^a) ^^'} '^^ BhM use the word order of that function, in the usual sense, to denote the number of factors of which each term consists. Thus, if any term were p^p^p^^ the oirder of the function would be r + 5 + ^. If the function be not homogeneous, the order of the function is as usual regulated by the order of the highest term. By the weight of a function we shall understand the sum of the suffixes attached to each factor. Thus, if any term were p^p^p^i the weight of the function would be r+25H-3#; or, again, if any term were PrP,Ptj this term would be of the third order, while its weight would be r + 5 + ^. In the case of every function, with which we shall be concerned, the weight will be the same for every term. THEIR ORDER AND WEIGHT. 59 67. On inspecting the expressions given above for «j, «,, 5g, &c. in terms of the coefficients, it is obvious that the weight of every term. in s^ is two, in ^3 is three, and it is easy to conclude by induction that the weight of every term in s^ is w. In like manner, it is evident that the weight of Sa"^' is m +^, of 2a*")8^7' is w + p + y, &c. This may be proved in general as follows: If for every root a, /8, 7, &c. we substitute X times a, \ times /8, X times 7, &c., we evidently multiply the function 2a"')8^7* by A""^". But it is known that if we multiply every root by \, we multiply Pt by X, p^ by V, p^ by X', &c. It follows then that Sa"')^7' expressed in terms of the coefficients must be such that if we substitute for f>,, X^„ for p^^ \^p^y and so on, we shall multiply every term by X"***^ ; and this, in other words, is saying that the weight of every term is w + /> + j. 58. Since p^ = a + /3 + y^^' &c., p, = a (/8 + 7 + &c.) + /87 + &c., &c., and none of the coefficients, ^3, p^^ &c. contains any power of a beyond the first, it is plain that the order of any symmetric function ^oTff'y^ (where m is supposed to be greater than p or q) must be at least m. For, of course, unless there are ^t least m factors, each containing a, oT cannot appear in the pro- duct. But, conversely, any symmetric function, whose order is «», will contain some terms involving a". For if q^^ q^^ q^^ &c. be the sum, sum of products in pairs, in threes, &c. of ^, 7, 8, &c., we have p^ = a + g'„ p^ = cnq^ + q^^ p^ = aq^ + g'3, &c., and the coefficient of the highest power of a in such a term ^^ p^p^p^^ will be qi[q^q^ ; and, conversely, the multiplier q^q^q* can only arise from the term p^p^pl* It therefore cannot be made to ▼anish by the addition of other terms. It follows then that the order of any symmetric function 2a"'y8^7* is equal to the greatest of the numbers w, ^, q\ for we have proved that it cannot be less than that number, and that it cannot be greater, since functions of a higher order would contain higher powers of a than a"". By the help of the two principles just proved (that the weight is the degree in the roots, and the order the highest 60 SYMMETRIC FUNCTIONS. degree in any one root), we can write down the literal part of any symmetric function, and it only remains to determine ttie coefficients. Thus if it were required to form 2a* {0 - 7)*, we see on inspection that this is a function whose weight is four, and that it is of the second order ; that is to say, there cannot be more than two factors in any term. The only terms then that can enter into such a function are p^j P^Px^ P%i *°^ ^^^ calculation would be complete if we knew with what coefficients these terms are to be aiSfected.* 59. Symmetric functions of the differences of the roots of equationsf being those with which we shall have most to deal, it may not be amiss to give a theorem by which the sum of any powers of the differences can be expressed in terms of the sums of the powers of the roots of the given equation. Expanding (x^a)^ by the binomial theorem, and adding the similar expansions for {x — /8)"*, &c., we have at once S (a? - a)"* = «^aj- - w«,a?**-* + im (m - 1) 5,a;"^ - i&c. Now if we substitute a for x in 2 (a? — a)"* it becomes (a — i8)"*+(a-7)*' + &c. ; similarly if we substitute 13 for x it becomes {ff — a)** + (^ - 7)** + &c., and so on ; and when we add the results of all these substitutions, if m be odd, the sum vanishes, since the terms (a — /S)*", (^ — a)*" cancel each other. If 7n be even, the result is 22 (a — /8)"*. But when the same substitiitions are made on the right-hand side of the equation last written, and the results added together, we get If m be odd, the last term will be —5^5^, which will cancel the first term, and, in like manner, all the other terms will destroy each other. But if m be even, the last term will be identical with the first, and so on, and the equation will be divisible by two. Thus, then, when m is even, we have 2 (a- ^r = Vm- ^'^iVi + i^ ('w- 1) VmHi-&c-, * The foregoing example of the calculation of Lafiy, £a</3, £a*, in eflEect showi how we can in eveiy case obtain for the determination of the ooeffidentB the zeqidied number of linear relations. f Such functions have been called critical functions. PARTIAL DIFFERENTIAL EQUATION. 61 where the coefficients are those of the binomial until we come to the middle term with which we stop, and which must be divided by two. Thus 60. Any function of the differences will of course be un- changed if we increase or diminish all the roots by the same quantities, as, for instance, if we substitute a;— X for a; in the given equation. It then becomes «"- {p, + n\)x'^ 4- {ft + (n- 1) \j>, + ln (n- 1) X»} x"^ - {ft + (w- 2) Xft + &C.1 iB*^ + &c. = 0. Now any function ^ of the coefficients ^,, ft, &c. will, when we alter ft into ft + Sp^j ft into ft + 8p^, &c., become It then, in any function of p^^p^^ &c., we substitute />, + n\ for ft, ft + (n ~ 1) \p^ + ^w (n — 1) X" for ft, &c., and arrange the result according to the powers of X, it becomes But since we have seen that any function of the differences is unchanged by the substitution, no matter how small X be, it is necessary that any function of the differences, when expressed in terms of the coefficients, should satisfy the differential equation Ex. 1. Let it be required to form T (a — /3)*. We know that its order and weight are both = 2. It must therefore be of the form Ap^ + Bpi^, Applying the differential equation, we have {(n —1)A + 2nB} p, = 0, whence B is proportional to n - 1 and ^ to - 2». The function then can only differ by a factor from (» - 1) p,* - 2np^ The factor may be shewn to be um'ty by supposing a = 1 and all the other roots = 0, when p^ = !,/>, = 0, and the value just written reduces to n - 1, as it ought to do. Ex. 2. To form for a cubic the product of the squares of the differences (o - /5)' (/3 - yY (y - a)«. This is a function of the order 4 and weight 6. It must 62 SYMMETRIC FUNCTIONS, theief oie be of the form Operating with 3 -z — h 2i>. -=- + »- -j— it becomes dp^ "^^dp^ ^^dp^ (2ii + ZB) p^p^ 4- (25 + 9C) p^p^^ 4- (-B + 6i) + 6^ p^^p^ + (C 4- 4^0 i'sPi", and as this is to vanish identically, we must have C- — 4iE, B = 18-E, -4 = — 27-E, J) = — 4^, or the function can only diflEer by a factor from PiW + 18/>ii>.i>3 - 4i>2« - ^PzPi^ - "^Pz^- The factor may be shewn to be unity by supposing y and consequently/), to be = 0. 61. We shall in future usually employ homogeneous equa- tions. Thus, writing x\y for a:, and clearing of fractions, the equation we have used becomes «" - p.x'^'y •Vp^x^'^f. . .± py = 0. We give a;" a coefficient for the sake of symmetry ; and we find it convenient to give the terms the same coefficients as in the binomial theorem ; and so write the equation or, as this may be for shortness represented, One advantage of using the binomial coefficients is, that thus all functions of the differences of the roots will, when expressed in terms of the coefficients, be such that the sum of the numerical coefficients will be nothing. For we get the sum of the nume- rical coefficients by making a^r=:aj = ag = &c. = 1 ; but on this supposition all the roots of the original equation become equal, and all the differences vanish. When we speak of a symmetric function of the roots of the homogeneous equation, we understand that the equation having been divided by aj/"^ the corresponding symmetric function has d CL SB been formed of the coefficients — , — , &c. of the equation in - , and that it has been cleared of fractions by multiplying by the highest power of a^ in any denominator. In this way, every symmetric function will be' a homogeneous function of the co- efficients Oq, a^, &c. ; for, before it was cleared of fractions, it was HOMOQENEOUS EQUATIONS. 63 » a homogeneous function of the degree 0, and it remains homo* geneous when every term is multiplied by the same quantity. Or we may state the theory for the symmetric functions of the roots of the homogeneous equation, without first transforming it to an equation in x : y. If one of the roots of the latter equation be a, that is, if a factor of the function is a; — ay, then it is evident that the homogeneous equation is satisfied by any system of values x\ y' for which we have 33' = ay', since it is manifest that we are only concerned with the ratio x' : y\ And since the equation divided by y** is resolvable into factors, so the homogeneous equation is plainly reducible to a product of factors (a?y' — yx') {xy" — yx'') [xy'' — ya;'''), &c. Actually mul- tiplying and comparing with the original equation, we get fl^=yyy&c., wa,=-Sa:yy''&c., in(n-lJa,=SxVy &c., a« = ± a;V V &c., wa^., = T 2y VV &c. By making all the y's = 1, these expressions become the ordinary expressions for the coefficients of an equation in terms of its roots, x\ xf\ &c. And conversely, any symmetric function ex- pressed in the ordinary way in terms of the roots a?', x'\ may be reduced to the other form, by imagining each x' divided by the corresponding y', and then the whole multiplied by such a power of yy^' &c., as will clear it of fractions. Thus the sum of the squares of the differences 2 (a?' — a:'')* becomes 2 (ajy'-yVjy^y'*^ &c. And generally any function of the differences will consist of the sum of products of deter- minants of the form [xy'^ -y'x'') (x'y'^' -y'od") &c., by powers of y\ y\ &c. 62. The differential equation which we have given for functions of the differences of the roots requires to be modified when the equation has been written with binomial coefficients. Thus, if in the equation a^ + na^'^y + &c. = 0, we write a? + X for a:, the new a^ becomes a, + \a^^ a^ becomes a^ + 2\a^ + Va^, a becomes a^ 4 SXa,, H- i^\\ + \^a^^ &c., and any function ^ of the coefficients is altered by this substitution into ^ + x(.4n2a.^+3«,^^ + &c.) + &c 64 SYMMETRIC FaNCTIONS. Any fimction then of the differences, since it remains unaltered when x + \is substituted for x^ must satisfy the equation In like manner any function which remains unaltered, when y + X is substituted for y, must satisfy the equation Functions of the latter kind are functions of the differences of the reciprocals of the roots, and in the homogeneous notation consist of products of determinants of the form aj^ — ^a;", &c., by powers of sfj x'\ &c. Functions of the determinants oiyf' — y'di!' alone, and not multiplied by any powers of the aj's or the y's, will satisfy both the differential equations. 63. It is to be observed that the condition is not only necessary but sufficient, in order that <^ should be unaltered by the transformation a; + X (or x. We have seen that the coefficient of X in the transformed equation then vanishes, and the coefficient of X' is easily found to be d6 dd) dd> ^ \ ( d <Z o \' . «« d? + ^''' c?a. +««« 5a/ *"•+ 172 («• da/ 2«' 5S: + ^'^•j *' where, considering the second term as denoting y^^A.A^, the 00, a^, &c. which appear explicitly in A<^ are not to be differen- tiated. But this being so, the two terms together are = y^^ A • A0, where A . A^ denotes now the complete effect of the operation A upon A0. For, when we operate with the symbol on itself, the result will be the sum of the terms got by differentiating the a^, a,, &c., which appear explicitly, together with the result on the supposition that these a^, a,, &c., are constant. Thus, then A^ vanishing identically, we have A.A^ = 0, or the coefficient of X^ vanishes. So, in like manner, for the coefficients of the other powers of X. PABTIAL DIFFERENTIAL EQUATIONS. 65 Ex. To fonn for the cabic a^ + SajaE^y + ^a^n^ + a^^ the function This can be derived from Ex. 2, Art. 60, or else directly as follows. The fnnotion is to be of the order 4 and weight 6. Ic must therefore be of the form d d d, Operate with o, "TT "*" ^» ^ "** ^Sol » "^^ ^® ^ Equating separately to the coefficient of each term, and taking il = 1, we find J8 = -«, C=4, 2> = 4, -^ = -8. 64. M. Serret writes the operation a^ t + &c. in a compact form, which is sometimes conyenient. If we imagine a fictitious variable (f, of which the coefficients a^, a,, &€., are snch func« tions, that da^ da^ da^ - ■3r=''« 5f = ^"«' ^• = 3a., &c., then evidently -^^^^j- + 2«, ;t^ + Sa, -^^ + &c. In like manner na, -r^ + &c. may be briefly written -^ , where 17 is a variable, of which a^ a^, &c. are supposed to be such functions, that 65. The above operators may be represented by respectively, since the first of them operating on (a^, a,,..aJ(T, y)* produces the same effect as yS^, the second the same effect aa xh . If the function be expressed in terms of its roots K 66 ORDEfi AND WEIGHT OF ELIMINANTfl. then the two operators may be transformed into symbols operating on the roots, as we have {xSJ = - (a + iS + ..OaA. + a*8„ + ^^p +-, the proof of which may be supplied without diflSculty. LESSON VIIL ORDER AND WEIGHT OF ELIMINANTS. 66. When we are given k homogeneous equations in k variables (or, what comes to the same thing, k non-homoge- neous equations in Aj — 1 variables) it is always possible so to combine the equations as to obtain from them a single equation A = 0, in which these variables do not appear. We are then said to have eliminated the variables, and the quantity A is called the EUminant^ of the system of equations. Let us take the simplest example, that which we have already considered in the first lesson, where we are given two equations of the first degree aa; + J = 0, a'x -f 6' = 0. If we multiply the first equation by a\ and the second by a, and subtract the first equation from the second, we get aV — a!h = 0, and the quantity aV — a'b is the eliminant of the two equations. Now it will be observed, that we cannot draw the inference aV '-ab=^0 unless the two given equations are supposed to be simultaneous, that is to say, unless it is supposed that both can be satisfied by the same value of a?. For, evidently, when we combine two equations <^(a?) = 0, yfr {x) = 0, arid draw such an inference as Z0 [x) + myjr (a?) = 0, it is assumed that x means the same thing in both equations. It follows then that a6' — a'i = is the condition, that the two equations can be satisfied by the same value of a;, as may also be seen immediately by solving both equations for Xy and equating the resulting values. And so generally, if we * J^liminants are also called reiuUaTUs^ ELIMINANTS EXPRESSED BT THE ROOTS. 67 are given any number of equations, ?7= 0, V=^ 0, W= 0, &c.i we may proceed to combine them, and draw an inference such as IU+ m V+ n TF= 0, only if the variables have the same values in all the equations. And, if by combining the equations, we arrive at a result not containing the variables, this will vanish if the equations can be satisfied by a common system of values of the variables, and not otherwise. Hence for any such system of equations the eliminant may in general be defined as that function of the coeffidents^ whose vanishing expresses that the equa^ tions can be satisfied by a common system of values of the variables. 67. We have now to show how elimination can be performed, and what is the nature of the results arrived at. We commence with two equations written in the non-homogeneous form 03* - j,aj""^ + j^aj""" - &c. = 0, or ^ [x) = 0. The vanishing of the eliminant of these equations is, as we have seen, the condition that they should have a common root. If this be the case, some one of the roots of the first equation mnst satisfy the second. Let the roots of the first equation be a, ^, 7, &c., and let us substitute these values successively in the second equation, then some one of the results '^(a), '^ (/3), &c. must vanish, and therefore the product of all must certainly vanish. But this product is a symmetric func- tion of the roots of the first equation, and therefore can be expressed in terms of its coefficients, in which state it is the eliminant required. The rule then for elimination by this method, is to take the m factors Vr (a) = a* - 2,a"-* + j^a""* - &c., t (7) = 7'' - 2^ + 2,7'^ - &c., &c., to multiply all together, and then substitute for the symmetric functions (a^7)", &c., their values in terms of the coefficients of the first equation. Ex. To eliminate x between a* — p^x + p, = 0, a* — q^x + ^^j = 0. Multiplying (a« - fta + ^2) (/3« - ji^ + J,), we get ««^ - ^,a/3 (a + ^) + ft (a2 + /3«) + q^^a^ - ftft (« + « + ft* i 68 ORDER AND WEIGHT OF ELIMINANTS. and then sabatitating a + /9 = Pi, a/3 = />» a* •{• p^ = Pi* - 2p^ we have !>»• -PiPrfi + ft (Pi* - 2?,) +l>rfi' - MiPi + ft*. «f (i^ -?«)• + (Pi - 5i) (i>i?j -JPrfi)» which is the eliminant xeqnized. 68. We obtain in this way the same result (or at most results differing only in sign), whether we substitute the roots of the first equation in the second, or those of the second in the first. In other words, if a', ^, y, &c. be the roots of the second equation, the eliminant may be written at pleasure as the continued product of (j) (a'), 4> (1^)} ^ (tOj ^^n ^^ ^ ^^ product of -^(a), -^{^8), "^(t), &c. For remembering that ^ [x) = (» — a) (a: — /5) {x - 7), &c., the first form is (a' - a) (a' - /8) (a' - y) i&c. {^ - a) (/S' - /3) [^ - y) &c, and the second is (a-*') (a-/8') («-7') &c. (/3- a') (/3- /8') (^3-^) &c In either case we get the product of all possible differences between a root of the first equation and a root of the second; and the two products can at most differ in sign. 69. If the equations had been given in the homogeneous form, with or without binomial coefficients, a^x"^ + ma^oT'^y + \m {m - 1) a^x'^f + &c. = 0, J^a;" + wJ,JB"'\y + iw (n - 1) b^^^t/^ + &c. = 0, V we can reduce them to the preceding form bj dividing them re- spectively by aj/^^ J^", when we have^, = ^, ?i = — -r-' , &c. We substitute then these values for p^^ q^^ &c. in the result obtained by the method of the last article, and then clear of fractions by multiplying by the highest power of a^, or \ in any denominator. Thus the eliminant of a^x^-k-^a^xy^aj^^ \a? -f 26^0:;^ + hj/^^ obtained in this manner from the result of Ex., Art. 67, is [a^\ - a,\Y -f 4 [aj>^ - a,ij (a, J, - aj>,). OF TWO HOMOGENEOUS EQUATIONS. 69 It is evident thus that the eltminant is always a homogeneous /unction of the coefficients of each equation. For before we cleared of fractions^ it was evidently a homogeneous function of the degree 0, and it remains homogeneous when every term is multiplied by the same quantity. The same thing may be seen by applying to the equations directly the process of Art. 67. Let the values which satisfy the first equation be x^i/'j ^''y'\ &c. ; then, if the equations have a common factor, some one of these values must satisfy the second equation. We must then multiply together (J.oj'" + nJ^x'-y + &c.) ( J,*''" + nh^x'^^^y'' + &c.) (&c.), which is a homogeneous function of the coefficients \^ h^^ ... of the second equation, and of course remains so after substituting for the symmetric functions (a;V &c.)** &c. their values in terms of the coefficients of the first equation. And in the same manner the function is homogeneous as regards the coefficients of the ■first equation. 70. The eliminant of two equations of the m^ and n** orders respectively^ is of the n^ order in the coefficients of the first equor tionj and of the tw** in the coefficients of the second. For it may be written either as the product of m factors ^ (oi), ^ (/3), &c., each containing the coefficients of the second equation in the first degree, or else as the product of n factors (a^), (/S^), &c., each containing in the first degree the coeffi- cients of the first equation. Or confining our attention to the form '^(a).'^(i8).&c. we can see that this form, which obviously contains the coefficients of the second equation in the degree 9n, contains those of the first in the degree n, since the sym- metric functions which occur in it may contain the n**, and no higher, power of any root (Art. 58). 71. The weight of the eliminant is mn; that is to say, the sum of the suffixes in every term is constant and ^^mn. For if each of the roots a, fi ; a\ fi'j &c. be multiplied by the same factor X, then since each of the mn differences a — a^ (see 70 OKDER AND WEIGHT OP ELIMTNANTS. Art. 68) Is multiplied by this factor \, the eliminant will be multiplied by \*"**. But the roots of the two equations will be multiplied by \ if for jp^j q^ we substitute Xp,, \q^] ft>r^„ y,; X^Pj, X*Jj ; &c. We see then, that if we make this substitution in the eliminant, the eflFect will be that every term will. be multiplied by X*"** ; or, in other words, the sum of the suffixes in every term will be mn. The same thing may also be seen to follow from the principle of Art. 57. In yjr {x) the sum of the index of every term and the suffix of the corresponding coefficient is n ; that is to say, '^ {x) consists of the sum of a number of terms, each of the form q^^gX*. If, then, we take any term at random in each of the factors -^ (a), yjr (^), &c., the corresponding term in the product will be q^^iq^jq^fi^^^^^j &c., and if we combine with this all other terms in which the same co- efficients of the second equation occur, we get ?„_,g'„_i2„»4Sa*i8'7*, &c. The sum of the suffixes of the g's is w— V+n— y+n— ^+&c., or since there are m factors, the sum is ?wtt— (i+y+ ^ + &c.). But, by Art. 57, the sum of the suffixes of the p's in the ex- pression for 2a'^7*, &c. is i-f-J + A; + &c. Therefore the sum of both sets of suffixes is mriy which was to be proved. The result at which we have arrived may be otherwise stated thus:* If p^^ 2i (contain any new variable z in the first degree ; if Pa ?8 ^''^^'w i^ ^'^ ihe second and lower degrees; if p^^ j, in the thirdy and so on ; then the eliminant will in general contain this variable in the mn*^ degree. It is evident that the results of this and of the last article are equally true if the equations had been written in the homo- geneous form a^a;"* + &c., because the suffixes in the two forms mutually correspond. And again, from symmetry, it follows that the result of this article would be equally true if the equations had been written in the form a^x"^ + ma^^^^y + &c., where the suffix of any coefficient corresponds to the power of x which it multiplies, instead of to the power of y, ^■^■^^*^'^^^^^^^'^-^^^^^— ^ III ■ I ■■-^— ^ I ■ _ l^l■-■^■■■l■^■■■^■ p^^.^— ■ i ^B^i^— ■^^■^M^^M^M^— — ^ * Or again thus : if in the eliminant we substitute for each coefficient p^, the tenn «* which it multiplies in the original equation, every term of the eliminant wiU be divisible by a;"»». Or, in the homogeneous form, if we substitute for each coefficient Aa the term x*y^*, which it multiplies, every term of the eliminant will be divisible by ar^wyww. COMMON VALUES IN THREE VARIABLES. 71 72. Since the eliminant is a function of the differences between a root of one equation and a root of the other, it will be unaltered if the roots of each equation be increased by the same quantity; that is to say, if we substitute a; + \ for a: in each equation. It follows then, as in Art. 60, that the elimi- nant must satisfy the differential equation d^ , ^, c7A - dA f ^. dS f, dp, "^'dp^ dq^ '^'dq^ or, as in Art. 62, if the equations had been written with binomial coefficients, we have dA ^ dA f, . dA ^, dA f. 73. Given two homogeneous equations between three variable^^ of the w** and n** degrees respectively^ the number of systems of values of the variables which can be found to satisfy simultaneously ike two equations is mn,^ Let the two equations, arranged according to powers of x^ be ax"^ + [by + cz) a;"*"' + [dy^ + eyz + /«") a;"*"* + &c. = 0, aV + (b'y + cz) a;"-' + [dy + e'yz +fz^) a?""* + &c. = 0. If now we eliminate x between these equations, since the co- efficient of x^~^ is a homogeneous function of y and z of the first degree, that of a?"*^ is a similar function of the second degree, and so on, — it follows from the last Article that the eliminant will be a homogeneous function of y and z of the wn** degree. It follows then that mn values of y and z'f can be found which will make the eliminant = 0. If we substitute any one of these in the given equations, they will now have a common root when ♦ These equations may be considered as representing two curves of the m** and n* degrees respectively; the geometrical interpretation of the proposition of this Article being, that two such curves intersect in mn points. The equations are re- duced to ordinary Cartesian equations by making e = 1. t The reader will remember that when we use homogeneous equations, the ratio of the variables in all with which we are concerned. Thus here, z may be taken arbitraiily, the coneBponding value of y being determined by the equation in y : jr. 72 ORDER AND WEIGHT OP ELIMINANTS. solved for x (since their eliminaut vanishes] ; and this value of Xj combined with the values of y and z already found, gives one system of values satisfying the given equations. So we plainly have in all mn such systems of values. We shall, in Lesson x., give a method by which, when two equations have a common root, that common root can immediately be found. Ex. To find the coordinates of the four points of intersection of the two conies aac^ + by' + cz'* + 2/y« + 2gzx + 2hxy = 0, a V + by + c'z' + 2/'y« + S/aa? + Vt*xy = 0. Arrange the equations according to the powers of ar, and eliminate that yaiiable, as in Art. 67; then the result is {(o^O y« + 2 (a/') yz + [px^ 2«}« + 4 [(aA')y + (a^') «] [(*A')y»+ {(V) + 2 (/^Oly'^ + fCcAO + s (/^')}2'y+(<»02^ = 0, where, as in Lesson I., we have written {ab*) for ab* - a*b. This equation, solred for y : z, determines the values corresponding to the four points of intersection. Having found these, by substituting any one of them in both equations, and finding their common root, we obtain the corresponding value oi x\z. We might have at once got the four values of a; : e by eliminating y between the equations, but substitution in the equations is necessaiy in order to find which value of y corresponds to each value of X, By making z = 1, what has been said is translated into the lang^nage of ordinaiy Cartesian coordinates. 74. Any symmetric functions of the mn values which simul" ianeously satisfy the two equations can be expressed in terms of the coefficients of those equations. In order to be more easily understood, we first consider non-homogeneous equations in two variables. Then it is plain enough that we can so express symmetric functions involving either variable alone. For, eliminating y, we have an equation in Xj in terms of whose coefficients can be expressed all sym- metric functions of the mn values of x which satisfy both equa- tions. Similarly for y. Thus, for example, in the case of two conies, xy^ &c. being the coordinates of their points of intersec- tion, we see at once how to express such symmetric functions aa X, + X, + aj,, + a?,,, y\ + y\ + y\ + y\j &c., and the only thing requiring explanation is bow to express sym- metric functions into which both variables enter, such as To do this, we introduce a new variable, t = 7ix + fiy^ and by the help of this assumed equation eliminate both x and y from the given equations. Thus y Is immediately eliminated by substi- OF THHEE HOMOGENEOUS EQUATIONS. 73 tntmg in both its value derived from tss\x + fAy^ and then we have two equations of the m^ and n*^ degrees in x, the eliminant of which will be of the mn*^ degree in f, and its roots will be obviously Xa;, + /Ay,, \a?,, + /Lty,,, &c., where x^y^^ x^^y^^ are the values of x and y common to the two equations. Tbe coeffi- cients of this equation in t will of course involve X and §1. We next form the sum of the A^ powers of the roots of this equa- tion in tj which must plainly be = lXa;, + A*y,)*+(Xa?,,+/MyJ*+&c» The coefficient, then, of X* in this sum will be 2a;/ : Uie coeffi- cient of X*"*/A gives us 'S.x^^y^^ and so on. Little need be said in order to translate the above into the language of homogeneous equations. We see at once how to form symmetric functions involving two variables only, such as 2yAA,A^» for these are found, as explained. Art. 61, from the homogeneous equation obtained on eliminating the remaining variable ; the only thing requiring explanation is how to form symmetric functions involving all these variables, and this ia done precisely as above, by substituting < = Xa? + iiy. Ex. To form the sjmmetric functions of the cooidinatea of the four points common to two conies. The equation in the last Example givea at once yjy.,yu;y^ = (a^O* + * {a^n (<?/) ; ^^^..^/r = (a*? + 4 (ah') {hh*) ; and, from symmetry, xfl,fl„jK,^ = {fxff + 4 ifif) {ef), - ^ (y.yuy./^J = 4 {(o^) iqf) + {ah') (eg') + (ag') (ch') + 2 (a^') {fg')], <kc. To take an example of a function iuTolying three variables, let us form ^ (^;yAA,A^\ which oorvespondfl to £ (x'y*) when the equations are written in the non-homogeneoua form. By the preceding theory we are to eliminate between the given equations, and t = \x + fiy, and the required function will be half the coefficient of \/i in £ (<>/^^<,^;sV)* If the result of elimination be At^ + (B\ + Cn) t^z + (m? + Ekyi + Fy?) i^z'^ + <kc., 2 («»/2 V*./V) = (B\ + Cm)« - 2 A (D\^ + ^M + Ffi^, and 2 (x;yAA./^^Ji ^BC-AE. By actual elimination A = (oJO* + 4 (aV) (bh'), B = 4: {(ba') (bg^ + (bf) (ah') + (bk') (af) + 2 (bh') (gh')], (7 = 4 {(dbT) (af) -{-(asn (bh') + (ah') (bf) + 2 (ah') [fh')} J? = 4 {(acO (bh') + (*0 (ah') - 2 (af) (hf) - 2 (bg^ (hg^ + 4 (hf) (kg')]. 75. To form the eliminant of three homogeneous equations in three variables of the «i**, n**, and p^ degrees respectively. The vanishing of the eliminant is the condition that a system L 74 ORDER AND WEIGHT OF ELTMINANTS. of values of x^ y, z can be found to satisfy all three equations.^ When this is the case, if we solve from any two of the equations, and substitute successively in the remaining one the values so found for a?, y, ^, some one of these sets of values must satisfy that equation, and therefore the product of all the results of substitution must vanish. Let J*', y', «'; x^\ t/\ «", &c. be the systems of values which satisfy the last two equations, which (Art. 73) are np in number: substitute these values in the first, and multiply together the np results ^ (a?', y', /), 4> [x'\ y, O) &c- The product will plainly involve only sym- metric functions of x\ y\ z\ &c., which (Art. 74) can all be expressed in terms of the coefficients of the last two equations ; and, when they are so expressed, it is the eliminant required. 76. The eliminant is a homogeneous function of the np^ order in the coefficients of the first equation ; of the mp*^ in those of the second ; and of the mn*^ in those of the third. For, each of the np factors 4> {^% ^> ^') is a homogeneous function of the first degree in the coefficients of the first equation ; and the expression of the symmetric functions in terms of the coefficients only involves coefficients of the last two equations, from solving which x\ y\ z\ &c. were obtained. The eliminant is therefore of the r?p** degree in the coefficients of the first equation; and in like manner its degree in the coefficients of the others may be inferred. 77. The weight of the eliminant will he mnp ; that is to say, If all the coefficients in the equations which multiply the first power of one of the variables^ z^ he affected with a suffix 1, those which multiply z^ with a suffix 2, and so on; the sum of all the suffixes in each term of the eliminant will be equal to mnp. In other words: If all the coefficients which multiply z contain a new variable in the first degree ; — if those which multiply z^ contain it in the second and lower degrees^ and so on; then the eliminant will contain this variable in the degree mnp. *. If the three equations represent curves, the vanishing of the eliminant is the condition that all three curves shall pass through a common point. COMMON VALUES IN THREE VARIABLES, 75 This is proved as in Art. 71. In the first place, it is evident that if a homogeneous equation of the 'rrl^ degree be satisfied bj values a?', y\ z' ; and if the equation be altered by multiplying each coefficient by a power of \, equal to the power of «, which the coefficient multiplies, then the equation so transformed will be satisfied by the values Xa;', X/, z' ; or, in general, that the result of substituting Xa*^, \y^, / in the transformed equation is X"* times the result of substituting x\ y', z' in the untransformed. Thus, take the equation a3*+y' — «'-2'aj — «y, the transformed, is 0?" + y* - XV — XViB - X«y' ; and, obviously, the result of sub- stituting Xo;', Xy', z' in the second is X' times the result of substituting x\ y, z' in the given equation. If, then, the three given equations be all transformed by multiplying each coeffi- cient by a power of X equal to the power of z^ which the coefficient multiplies, it follows, if x\ y, z' be one of the systems of values which satisfy the two last of the original equations, that the transformed equations will be satisfied by (Xo;^, \y\ z\ and the result of substituting these values in the first will be X"*^ (a^, y, «'). The eliminant, then, which is the product of np factors of the form ^ (a?', y', «') will be multiplied by X*^. If, then, any term in the eliminant be «^V«» &c., where the suffix corresponds to the power of z^ which the coefficient multiplies, since the alteration of a^ into X*a^, h^ into X% &c., multiplies the term by X**"', we must have h + ?+ &c. «= mnp. Q.E.D. 78. It is proved, in like manner, that three equations are in general satisfied by mnp common values; that any symmetric function of these values can be expressed in terms of their co- efficients ; and that we can form the eliminant of four equations by solving from any three of them, substituting successively in the fourth each of the systems of values so found, forming the product of the results of substitution, and then, by the method of symmetric functions, expressing the product in terms of the coefficients of the equations. In this way we can form the eliminant of any number of equations ; and we have the follow- : ing general theorems : The eliminant of k equations in k—l, independent variables is a homogeneous function of the coefficients 76 EXPRFSSIOK OF ELIMINANTS AS DETERMINANTS. of each equation^ whose order is equal to the product of the degrees of all the remaining equations. If each coefficient in all the equations he affected with a suffix equal to the power of any one variable which it multiplies^ then the sum of the suffixes in every term of the eliminant will he equal to the product of the degrees of all the equations. And, again, if we are given k equations in k variables^ the number of systems of common values of iJie variables^ which can be found to satisfy all the equations^ mil h^ equal to the product of th^ orders of the equations* LESSON IX. EZPBE9SI0N OF ELDflHAKTS AS DETEBHINAirrS. 79. The method of elimination by symmetric fimctions is, in a theoretical point of view, perhaps preferable to any other, it being nniversally applicable to equations in any number of variables ; yet as (in the absence of tables of symmetric func* tions) it is not very expeditious in practice, and does not yield its results in the most convenient form, we shall in this Lesson give an account of some other methods of elimi-* nation. The following is the method which most obviously presents itself. It is in substance identical with what is called elimination by the process of finding the greatest common measure. We have already seep that the eliminant of two linear equations aa; + i = 0, a'x + V^O is the determinant oJ' — ha' = 0. If now we have two quadratic equations ax* + bx + c = Oj aV + b'x + <;' = 0, multiplying the first by a\ the second by ay ^nd subtracting, we get {ah') X + [ac') = ; and, again, multiplying the first by c\ the secoud by c, 8ob« firacting, and dividing by a;, we get (ac') X + {he') = 0. eulbr's method. 77 The problem is now reduced to elimination between two linear equations, and the result is 80. So, again, if we have two cubic equations we multiply the first bj a\ the second by a, and subtract ; and also multiply the first by d\ the second by d^ subtract and divide by x. The problem is thus reduced to elimination be- tween the two quadratics {aV) a?-^(ad)X'{' {ad') = 0, [ad') «" + ( Ji') x + [cd') = 0. By the last article the result is {[ad:Y'-(aV)[cd:)Y^-{[ad:){ac')-[^^ Now it is to be observed that the equation (ai') (of) + {ac') [db') + {ad:) [he') = is identically true. Consequently when we multiply out, the preceding result becomes divisible by (atf), and the reduced result is (ocf )» - 2 {ad:) {ah') {cdTj - {ad:) {ac') [hd:) + {acj {cd:) + {hd:f [aV) - {aV) {he') (cef) = 0. The reason that in this process the irrelevant factor (ocT) is in- troduced is that, \iact = a'd^ and therefore a to a' in the same ratio as c2 to (f , we must get results differing only by a factor, if from the first equation multiplied by a' we subtract the second equation multiplied by a, or, if from the first equation multiplied by tf , we subtract the second equation multiplied by d. Thus, on the supposition (o^) := 0, even though the original two cubics have not a common factor, the two quadratics to which we reduce them would have a common factor. In general then, when we eliminate by this process, irrelevant factors are intro- duced, and therefore other methods are preferable. 81. Euler^a Method. If two equations of the w** and n* degrees respectively have a common factor of the first degree, we must obtain identical results, whether we multiply the first 78 EXPRESSION OF ELIMTNANTS AS DETERMINANTS. equation by the remaining n — l factors of the second, or the second by the remaining m — 1 factors of the first. If then we multiply the first by an arbitrary function of the (n -1)** degree, which, of course, introduces n arbitrary constants ; if we multiply the second by an arbitrary function of the (m — 1)** degree, intro- ducing thus m constants ; and if we then equate, term by term, the two equations of the (tw + w — 1)** degree so formed, we shall haye m + n equations, from which we can eliminate the m + n introduced constants, which all enter into those equations only in the first degree ; and we thus obtain, in the form of a deter- minant, the eliminant of the two given equations. Ex. To eUminate between aas* + bxy + cy» = 0, aV + b'xy + cy = 0. We are to equate, tenn by term, (Ax + By) (oa* + day + <?y«) and {A'x + B'y) (a'a? + Vxy + <?y). The four resulting equations are Aa - A'a' = 0, Ae+Bh- AW - BV = 0, . Be - B^o' = 0, from which, eliminating A^ B, A*, B*, the result is the determinant «» 0, a% *, «» v, a' ^f *, ^, V 0, c, 0, if 82. This method may be extended to find the conditions that the equations should have two common factors. In this case it is evident, in like manner, that we shall obtain the same result whether we multiply the first by the remaining w — 2 factors of the second, or the second by the remaining m - 2 factors of the first. As before, then, we multiply the first by an arbitrary function of the n — 2 degree (introducing 9i — 1 constants), and the second by an arbitrary function of the m - 2 degree ; and equating, term by term, the two equations of the m + n --2 degree so found, we have ?n + n — 1 equations, from any r/i 4 n — 2 of which, eliminating the m + n — 2 introduced constants, we obtain m-\-n — l conditions, equivalent, of course, to only two independent conditions. stlvester's method. ^9 Ex. To find the oonditionB that aa» + ba^ + exy^ + <^ = 0, a'«» + b's^ + f^xt^ + cfy» = 0, should have two common factors. Equating ( J« + -By) (aa» + ba?y + ftry» + rfy«) = (^'« + Ry) [(^a^ + y*^ + c'ay* + rfy), we have Aa — A'af = 0, Ab-hBa^A'b' -B'a' = 0, Ad+Be-A'tt-B'e'-O, Bd - J^cf = 0, from which, eliminating A, By A\ B^^ we have the system of determinants [for the notation used, see Art. 3], a, by e, d, '. 0, a, by Cy d : a'y Vy <fy ^y 0, a'y Vy e'y cf = 0. 83. Sylvester^a dtalytic method. This method is identical in its results with Euler's, but simpler in its application, and more easily capable of being extended. Multiply the equation of the w** degree by aj"'\ al^'^y^ ^""V i &c. ; and the second equation by as"'"^ 35"'"*y» ^""'V) &Cm ^^^ we thus get wi + n equations, from which we can eliminate linearly the m 4 n quantities a?""^*^, oT^'^y^ a****'y7 &®') considered as independent un- knowns. Thus, in the case of two quadratics, multiply both by X and by y^ and we get the equations asi? + hai^y + cxy^ = 0, asi?y + Jary* + cy' = 0, clof^Vofy-^-do^^ =0, a Vy + J'«y' + c'y' = 0, from which, eliminating o^, o^y^ xx^^ y", we get the same deter- ^ minant as before, a, 5, c, a, &, c a , & , c a , & , c In general, it is evident by this method, that the eliminant is expressed as a determinant of which n rows contain the coeffi- cients of the first equation, and m rows contain the coefficients 80 EXPRESSION OP ELIMINANTS AS DETERMINANTS. of the second. Thus we obtain the rule already stated for the order of the eliminant in the coefficients of each equation. Ex. 1. Eliminate x between a + bx •{- ea^ = 0, «* = 1. Multiply the former by x, and pnt «* = 1, and we get tf + oaf + ^jB* = ; from this similarly b + ex + cui? = 0, and the eliminant is in determinant form and expanded = a» + 6» + tf» - Zabe. a, h, c = — a, bf e c, a, b bf Cf a bf Cf a e, a, b It can also be found by the method of symmetric functions : let w be an imaginaiy cube root of unity, then the roots of the second equation substituted in the first give the eliminant as (a + b + e) {a + b«» + eta*) (a + i«* + c«), which is thus the yalue of the determinant. Ex. 2. Similarly eliminating between a-hbx + csc^ + da? = 0, x* = 1, ttf bf Cf d df ttf bf c Cf df Of b bf Cf df a Of bf Cf d bf Cf df a Cf df a, b df cif bf c = {(a + c)2-(6 + d)«l{(a-c)« + (ft-rf)«} = (a + 6 + c + <Q(o — 6 + tf — d)(a + W — tf — eft)(a — W — tf + d%), Ex. 8. Generally, in a similar manner for w an imaginary nf^ root of unity, the determinant a, bf Cf ,»ml If Ctf Of tm»fC kf If (If bf Cf df ...a = (o + b + e +...+ 1) (a+ bu) +<?»«+...+ /««-') (a + ia>« + + ?»"-«) (a + *a>»-i + + ho). Ex. 4. Yarious determinants may be found by this method of elimination. Starting from the equations Oix + (i^ = Ofbjps+b^= 0, they are consistent if (oib^ = 0. Take the squares and product of these expressions, they also vanish together if the oiiginal equations be consistent ; hence, eliminating a^, xy, ^ dialy tically, a 1 ) 2aia2, «2^ must be a power of {afi^if and it is obviously (ajd,)'. Similarly {ajt^^ can be written down as a determinant of order 4. Again, {b^x + b^) {Cix + c^) = 0, {c^x + c^) {a^x + Ojy) = 0, {ciix + a^) {b^x + b^) — 0, are consistent if (^jc,) {CiCl^ (aj*,) vanish. Hence this product may be written b^Cif byC^ + Vi, V2 C,rtl, CjO^ + ^2^1, ^2^ afiif 0/2 + ^*11 ^2 bezodt's method. 81 And in like manner all determinants of the matrix «i'» *i*» *i*f Vif <?i«ii "ih iaiOff 26162, 2ciCf, biC^ + Vii Ci*h + <Vh> « A + ««*i Oj'» V» Ca'» *A, <V»» «A are prodncta of the third order of the qnantitiea (6|C^, (^iOs)f (<h^s)* 84. Bezouf» metJiod. This process also expresses the elimi- nant in the form of a determinant, but one which can be more rapidly calculated than the preceding. The general method will, perhaps, be better understood if we applj it first to the particular case of the two equations of the fourth degree Multiplying the first by a\ the second by a, and subtractingi the first term in each is eliminated, and the result, being diyLuble (ai') aj^+ (ac')«'y + MO ay' + (oe') y* = 0. Again, multiply the first by a'x^-Vy^ and the second by ax+by^ and the two first terms in each are eliminated, and the result, being divided by y*, gives {ac') »• + {{ad') + (i(/)} a?y + {(a/) + {hd')] xy" + {be') y" = 0. Next, multiply the first by aV + J'ay + c'l/' ; and the second by caf + bxy 4 cy^ ; subtract, and divide by y' ; when we get {ad') aJ» + {{ae') + {bd')] x^y + [{be') + {cd')] xy* + (ceO y" = 0. Lastly, multiply the first by a'a?'\^b'3i?y-{-c'x^'{-d'y*\ the second by aa? + bx^y + cxy* + dy* ; subtract, and divide by y* ; when we get (ae')aJ^ + {be') a?y'{'{ce') ajy' + ((^0 y" = 0. From the four equations thus formed, we can eliminate linearly the four qnantitiea a;', aj'y, xy*^ y", and obtain for our result the determinant {ab'), {ac') , {ad') , (oe') {ac'), (arf')+(Jc'), (ae') + (W) , (be') {ad'), {ae') +{b<y), {b/) -^ {cd'), {ce') (oeO, {be'), K), (<fo') 82 EXPRESSION OF ELIMINANTS AS ^DETERMINANTS. 85. The process here employed is so evidently applicable to any two equations, both of the w** degree, that it is unnecessary to make a formal statement of the general proof. On inspection of the determinant obtained in the last article, the law of its formation is apparent, and we can at once write down the deter- minant which is the eliminant between two equations of the fifth degree by simply continuing the series of terms, writing an (q/') after every (ae'), &c. Thus the eliminant is (ai'), K) ,(at?) ,(06-) ,(a/0 K), (a^)+(ief ), (a/)+(ie')+(af ), (J/)+(ceO , (<f) K) , (a/)+(ie'), (J/)+(c60 , {cf)Hd/), [df) («/), (*/), [cf), m^ef) It appears hence that in the eliminant every term must con- tain a or a^ ; as was evident beforehand, since if both of these were = 0, the equations would evidently have the common factor y = 0. It appears also that those terms which contain a or a' only in the first degree are {aV) multiplied by the eliminant of the equa- tions got by making a and a' = in the given equations. For every element in the determinant written above must contain a constituent from the first row, and also one from the first column ; but as all the constituents of the first row or column contain a or a', the only t erms which contain a and a' in only the first degree are [aV) multiplied by the corresponding minor ; and this, when a and a^ are made = 0, is the next lower eliminant. 86. It only remains to shew that the process here employed is applicable when the equations are of different dimensions; and, as before, we commence with the following example : Multiply the first equation by a^, the second by oo;', and subtract, when we have [la') ^ + [c<i') Q^y + {da') xf + (ea') y' = 0. In like manner, multiply the first by a'x-^Vy^ and the second by \ax + hy) x*j and we get K) x' + {{cV) + [da')} x'y + {[d¥) + {ea')} xy' + {eb') f = 0. caylby's statement. 83 This process can be carried no further ; but if we join to the two equations just obtained the two equations got by multi- plying the second of the original equations by x and by y, we have four equations from which to eliminate a?', x^y^ ay*, y". And in general, when the degrees of the equations are un- equal, m being the greater, it will be found that the process of Art. 84 gives us n equations of the {m - 1)** degree, each of these equations being of the first order in the coefficients of each equation : to which we are to add the m — n equations found by multiplying the second equation by a?"*"*"*, 0?"'*"*^, &c., and we can then eliminate the m quantities aj"*^, a?"''*y, &c., from the m equations we have formed. Every row of the determinant con- tains the coefficients of the second equation, but only n rows contain the coefficients of the first. The eliminant is, therefore, as it ought to be, of the n!^ degree in the coefficients of the first, and of the m^ in those of the second equation. 87. Cayletf^s statement of Bezouf 8 method. If two equations {xj y), '^ (fl?, y) have a common root, then it must be possible to satisfy any equation of the form -I- "Kyjr = 0, independently of any particular value of X. Take, then, the equation (a?, y) yjr (a?', /) - <l> {x\ y') yfr {x, y) = ; which, if and yfr have a common factor, can be satisfied inde- pendently of any particular values of x' and y\ We may in the first place divide it by xy^—yx\ which is obviously a factor: next equate to the coefficients of the several powers of x', y^ ; and then eliminate the powers of x and y as if they were independent variables, when the result comes out in precisely the same form as by the method of Art. 84. Ex. To eliminate between aa^ + bxy + ry* = 0, a'a» + d'xy + <?*y' = 0. when divided by «y' - ya/ gives {{ab')x+(ac^ y] x* + {(ocO «+ (30 y}y' = 0; and, eliminating Xy y between the coefficients of a! and y, s^Miratelj equaated to 0, we get the eliminant (cwO* + (^') (^) = <>• 88. We proceed now to the theory of functions of three variables, the eliminant of which, however, except in particular caseS| has not been expressed as a determinant, though it can 84 EXPRESSION OF ELIMINANT3 AS DETERMINANTS. always be expressed as tbe quotient of one determinant divided hj another. We shall shew, in the first place, how to form a function of great importatice in the theory of elimination. Given k equations in k variables, i« s 0, t; = 0, tc; = 0, &c., if we write u„ tt„ tt„ &c. fo^" j" I ^ ) 3" 1 &C'> *^®P ^^^ determinant &c. &c. &c. &c. is called Jacobi's determinant, or simply the Jacobian of the given equations, and will be denoted in what fpllows by the letter /. «i) ««> ^'a) ^1) ^iJ ^8) W^, tt?„ W,j 89. If any number of equations arp satisfied by a common system qfvalue^^ that system mil satisfy the Jacobian; and when the equations arp of the same degree^ it y)ill also satisfy the derived equations of the Jacobian with regard to each of the variables. The proof of this for three yariables applies in general. By the theorem of hpmogeneous functions we havp fCfij + yu^ + zuj^ = nw, xv^ +yv, +«V3 =n't?, Now if, as In Lesson ly., we write the minors of the Jacobian, obtained by suppressing the row and column containing u,, ^j, &c., U^j Vi} &c., tten if we solve these equations, we finfl (Art. 29) Jx = U^ni^ + F,w'v + ^FjW'V) frpna which it appiears at: once that if Uj Vf to vanish, J will vanish too. Again, differeptiatipg the eqpation just found, we have, for w' = n-^ = n, dJ dR dK dW, , „ „ „, dJ dU, dV^ dW, , „ Tr HT. ''dy^''''di-^''''l^^''^'df'^''^^-^^^ But remembering (Art. 27) that « ELIMINANT OF TEBNART EQUATIONS. 85 we see that the supposition t< = 0, i; = 0, t(; = (in consequence of which / is also = 0) makes -j- , -r- ^Iso to vanish. ax ay 90. We can now express as a determinant the eliminant of three equations, each of the second degree. For their Jacobian is of the third degree, and therefore its differentials are of the second. We have thus three new equations of the second degree, which ifiiX be also satisfied by any system of values common to the given equations. From the six equations, . Cm/ Utf Cfe/ 1. • 1 • • • then, u^VjWj-^ , i- , -r- , we can elimmate the six quantities a;', t^j z^^ yxj zx^ xy^ and so form the determinant required. Again, if the equations are all of the third degree, J is of the sixth, and its differentials of the fifth, and if we multiply each of the three given equations by a?', y', «*, yz^ zxj ary, we obtain eighteen equations, which, combined with the three differentials of the Jacobian, enable us to eliminate dialytically the twenty^ one quantities, x^^ x^y^ &c., which enter into an equatioi^ of the fifth degree. This process, howeveri cannot, without modifif» cation, be e:^tended further. Ex. 1. The Jacobian of t^ homogeneons equatioxis may likewise be employe^ to write dpfni in detenninant form their eliminante. Thms, for « iv^ + 2&ry + <?^ = 0, a* 7? + 26*jry + c^y« = 0, writing the Jacobiai^ a^ + Ta^xy + o^y*, the eliminant is a', h\ if Co, a„ Oa Ex. 2. For two cubica aafi + 2h:^ + Zcxf + <?y» = 0, a'»» + W^ + So'ay* + <?y = 0, the Jacobian may be written a^ + 4rtia:»y + 6ajSBy + 4a,ay» + a^^ and since its two derived functions yanish for a common yalue, the eliminant ia a^ h^ c^ d a\ h\ e, d' «o» «i» Oa* «s «l» «2> «w «4 86 EXPRESSION OF ELIMINANTS AS DETERMINANTS. («iV.)* = (*lCA)(<?l<^») W«3*») («lV») = Similarlj for two biquadratics with a like notation, the eliminant is df by Cf dj Cf I • a', y, e', d*, e', 0, a, (, e^ df e 0, a' *', c', <«', e' Oo, «!, Oj, OsJ «4> «5 «l» °7» flw «4» fl** «• Ex. 8. As in Ex. 4, Art. 83, we can easily determine the following and similar identities : «i*» «a*» fl»*» Sflja^ 2a^i, 2a,a, *i^ V, V, 2*A, 2&361, 26A <?!«, Cj^, c,2, 2<?jc„ 2c,<?i, 2tf,ci Vi> Vw V»> V» + *»C2» Vi + V» ^iCj + Vi <^i«i, 0202, <?ja8» ^8 + ^i^ht ^»®i + ^i^^w ^i**2 + <Vh ^A> <*»*» <*A» ^» + ''8^2» ^1 + <*A» **A + **«^i 6,C|, b^„ ijCj, 32<?, + ^8*^2, b^i + JjCj, ^itf, + h^i tfiOj, Cj^, CjOj, CjOa + Cs^hi ^^i + ^'i'** '^i*^ "*• *«*! a,6i, 02^2* ^^8» <^» + <*»^2> ^^l + ^J*t) ^1^2 + '*«*1 aydijO^d^ a^d^y a^^+a^d^ 03^1 + o,id^ Oid^-hit^i bid If b^f b^d^y b^d^ + b^d^ b^di + b^d^ b^d^ + b^ c^diy c^2t ^3^»» ^2^9 + ^'s^a <^8^1 + ^ A> <^ A + <V^ . 91. Dr. Sylvester has shewn that the eliminant can always be expressed as a determinant when the three equations are of the same degree. Let us take, for an example, three equa- tions of the fourth degree. Multiply each by the six terms (a?*, ajy, y", &c.) of an equation of the second degree [or gene- rally by the ^n (w — 1 ) terms of an equation of the degree (w— 2)]. We thus form eighteen [|w (w — 1)] equations. But since these equations, being now of the sixth [2n — 2] degree, consist of twenty-eight [n(2n — 1)] terms, we require ten [iw(n+l)] ad- ditional equations to enable us to eliminate dialytically all the powers of the variables. These equations are formed as follows : The first of the three given equations can be written in the form Ax* + Bt/-^ Czj the second and third in the form and the determinant {AB^C) which is of the sixth degree in the variables must obviously vanish for any values which satisfy all the given equations. We should form two similar determinants by decomposing the equations into the form Ay* + £x + Czj Az* + -Bo: + Cy. So again we might decompose ^yley's theory. 87 the equations into the forms Ao^ + %" + Cfe, -4 V + 5 V + G'z^ A"q^ •\-B''y*'\'G"z) for every term not divisible by a' or y" must be divisible by £) ; and then we obtain another deter- minant [AWC') which will vanish when the equations vanish together. There are six determinants of this form got by inter- changing 2*, y, and z in the rule for decomposing the equations. Lastly, decomposing into the form Adf + By^ + Cfe', &c., we get a single determinant, which, added to the nine equations already found, makes the ten required. In general, we decom- pose the equations into the form Aaf '\' By^ •\- Cz'' ^ such that a + i8 + 7 = n+2, and form the determinant [AB'C) ; and it can be very easily proved that the number of integer solutions of the equation a + i8 + 7 = n + 2 is Jn (w + 1), exactly the number required. 92. When the degrees of the equations are different, it is not possible to form in this way a determinant, which shall give the eliminant clear of extraneous factors. The reasou why such factors are introduced, and the method by which they are to be got rid of, will be understood from the following theoryj due to Prof. Cayley : Let us take for simplicity three equations, <t«, 17, w^ all of the second degree. If we attempt to eliminate dialytically by multiplying each by a?, y, «, we get nine equa- .tions, which are not sufficient to eliminate the ten quantities a:', aj'y, &c. Again, if we multiply each equation by the six quantities, ar', ary, y*, &c., we have eighteen equations, which are more than sufficient to eliminate the fifteen quantities x\ a:'y, &c. If we take at pleasure any fifteen of these equations, and form their determinant, we shall indeed have the eliminant, but it will be multiplied by an extraneous factor, since the determinant is of the fifteenth degree in the coefficients, while the eliminant is only of the twelfth (Art. 76, mn + wp + pm = 12, when m^n ^p = 2). The reason of this is, that the eighteen equations we have formed are not independent, but are connected by three linear relations. In fact, if we write the identity uv = vu^ and then replace the first u by its value, ax^ + by* + &c., and in like manner with the V on the right-hand side of the equation, we get ax\ + hy^v + cz\ + "ifyzv + &c. = a'aj'w + Uy'^u + &c. 88 EXPRESSION OF ELIMtNANTS AS DETEBHINANTS. ^H la like maniier, JVom the identities vw= wv, wu=uw, ve get ^H other identical relationa connecting the quantities a^u, y'w, a?Vf a'w, &c. The question then comes to this: "If there he tn+p linear equations in m variables, but these equations connected by P linear relationa so as to be equivalent only to m independent eqnations, how to express most simply the condition that all the equations can be made to vanish together." In the present case m=l5,p = 3. 93. Let us, for Bimplicity, take an example with numbera not quite so large, for instance, »( = 3,^ = 1. That la to aay, let us consider four equations, s, (, u, v, where a = a^x + h,y + c^z, t=a^x+hji+c^z, &c., these equations not being independent, but satisfying the relation, D^s-\- D^t + D^u-\- D^v = (i. Now I aay, in the first place, that if we form the determinant («,&,c,) of any three of these equations, s, f, u, this must contain D^ as a factor. For if i*, = 0, we shall have a, (, u connected by a linear rela- tion, so that any values which satisfied both s and t should satisfy « also; and therefore the supposition i>^ = would cause the determinant {<iji,c^ to vanish. And, in the second place, I Bay that we get the same result for, at most, one differing only in sign) whether we divide {a,h^o^ by D^ or {a,ft,c,) by D^ For {Art. 15) D^ (afi^c^) is the same as the determinant of which the first row is a,, 5,, c,, the second, a^, 6^, c,, and the third, J),ft^, -0,^,1 ^i'^tj but we may substitute for B^a^ ita value - D,a^- Djx^- D^a^, and in like manner for Dp^, D^^. The determinant would then (Art. 18) be resolvable into the auta of three others ; but two of these would vanish, having two rows the same, and there would remain B^ (0|i,c,) = — D^ ('*i^»c.}' It follows, then, that the eliminant of the system may be ex- pressed in any of the equivalent forms obtained by forming the determinant {a,\c^ of any three of the equations, and dividing by the remaining constant D^. Suppose now that we had five equations 9, /, w, «, w, con- nected by two linear relations Z*,s4-O,( + Z'jM + i?,« + i^,w = 0, £:,s + i:^t + E^u + E^v + E^io = 0. Eliminating w from these relations, we have [D,E^)3-^ {D^EJ t+[I}Ji^]u-i- lD,E^)v = 0, and we see, precisely as before, that the supposition (i>^£'J = REDUCED ELIMINANT. 89 would cause the determinant (aj}^c^) to vanish; and that we get the same result whether we divide (aj&,c,) by {D^E^) or divide the determinant of any other three of the equations by the complemental determinant answering to (DJE^)» This reasoning may be extended to any number of equations con- nected by any number of relations, and we are led to the following general rule for finding the elimlnant of the system in its simplest form. Write down the constants in the m+p equations, and complete them into a square form by adding the constants in the p relations ; thus 8] «i» *1) ^. D..K t] a„ J„ c^ A, ^. U] «8» ^» <^t A, ^. V] «4» Kl <^4 A, ^. to] «5> *61 ^6 A, E,, then the eliminant in its most reduced form is the determinant of any m rows of the left-hand or equation columns, divided by the determinant got by erasing these rows in the right-hand colunms. Thus, then, in the example of the last Article, we take the determinant of any fifteen of the equations, and, dividing it by a determinant formed with three of the relation rows, obtain the eliminant, which is of the twelfth degree, as it ought to be. 94. And, in general, given three equations of the m**, n**, and p^ degrees, we form a number of equations of the degree WI + W+/?- 2, by multiplying the first equation by all the terms oj""^*^, ^"^^V> ^^^ ^ ^^* We should in this manner have i(n4^-l)(w+p)4-^(;? + m-l)(^ + w) + ^(w + n-l)(»i+n) equations. But the number of terms, a?*"*"*'^, &c., to be elimi- nated from the equations formed, is J (m + w h-^ — 1) (m + n h-^), or, in general, less than the number of equations. But again, if we consider the identity uv = vuj which is of the degree m-\-nj and multiply it by the several terms af^j &c., we get i{p-l)p identical relations between the system of equations we have N 90 DETERMINATION OF COMMON BOOTS. formed ; and in like manner ^ (n — 1) w + J (w — 1) m other iden- tities ; and the number of identities subtracted from the number of equations leaves exactly the number of variables to be elimi- nated, and gives the eliminant in the right degree. 95. If we had four equations in four variables, we should pro- ceed in like manner, and it would be found then that the case would arise of our having m + n linear equations in m variables, these equations not being independent, but connected by n-hp relations; these latter relations again not being independent, but connected by p other relations. And in order to find the reduced eliminant of such a system, we should divide the deter- minant of any m of the equations by a quantity which is itself the quotient of two determinants. I think it needless to go into further details, but I thought it necessary to explain so much of the theory, the above being, as far as I know, the only general theory of the expression of elimlnants as determinants; since whenever, in the application of the dialytic method, any of the equations is multiplied by terms exceeding its own degree, we shall be sure to have a number of equations greater than the number of quantities which we want to eliminate. LESSON X. DETERMINATION OP COMMON ROOTa 96. When the eliminant of any number of equations vanishes, these equations can be satisfied by a common system of values, and we purpose in this Lesson to shew how that system of values can be found without actually solving the equations. The method is the same whatever be the number of the variables ; but for greater simplicity we commence with the system of two equations, ^ = 0, -^ = 0, where f = hy + b^,y + b„.y + &c. = 0. BY DIFFERENTIAL COEFFICIENTS OF ELIMINANT. 91 Let US suppose that some root of the second equation, x = a satisfies the first, and therefore that It the eliminant of the system vanishes. Now in (f) we may alter the coefficients [a^ into «„+-4^i a^^ into a^^^ + A .„ &c. ); and the transformed equation a^x"^ + ««rn«^"*"' + &<5- + ^.M^^"* + ^w-i^^*""' + &c. = will obviously still be satisfied by the value a? = a, provided only that the increments -4^, -4„_i> &c. are connected by the single relation ^«a" + ^..ta""' + &c. = 0, since the remaining part of the equation, by hypothesis, vanishes for x = a. The transformed equation then has a root common with '^, and therefore the eliminant between ^ and that trans- formed equation vanishes. But this eliminant is obtained from -iB, the eliminant of ^ and ^, by altering in it a^ into ««,+-4^) &c. The eliminant so transformed is We have £ = by hypothesis ; and since the increments A^^ &o. may be as small as we please, the terms containing the first powers of these increments must vanish separately. We have then A^ -^ — h A^, -^ 1- &c. =» 0. This relation must be iden- tical with the relation A^jx^ + A^^oT'^ + &c. = 0, which we have seen is the only relation that the increments need satisfy. It follows then that the several differential coefficients are pro- portional to a**, a*""*, &c., and therefore that a can be found by taking the quotient of any two consecutive differential co- efficients. Cob. 1. If a^^ a^ be any two coefficients in ^, we must have, . — » dR dR dR dR . « . i. when ^ = 0, 3- : 3 — t: -r~ • -3 — 1 smce the quotient as well dap da i^ dag dag_^ of the first by the second as of the third by the fourth will = a*. It follows that 3- -J -T- -J — vanishes when 5 = 0, and dap dag_^ dag da ^ therefore must contain i2 as a factor; or, in other words, dR dR dR dR • -n t* •/> T- J 3 — T- contains 5 as a factor if we have o + o' — r + «, dapdaq dOfdas 92 DETERMINATION OF COMMON ROOTS. Cor. 2. It b evident, by parity of reasoning, that the differ- ential coefficients of the eliminant, with regard to the several coefficients in ^, are proportional to a*, a^'^ &c. ; and hence, as in the last corollary, that, when 5 = 0, -7— : -= — :; -^r • -77- « ,_ ,^ , ' ,^ ' ' dam dd X db^ db^ * , dR dR dRdR . ^ % '"* , ^ V"* or that J- 31 — -J- jT contains 5 as a factor when we have da dbg cCttr ab^ p + q^r-^a. GoR. 3. Or, again, if we substitute in the second equation the values of a", a**"\ &c. given above, we have , dR , dR . p . 'n-X when JS = 0. But the left-hand side of this equation cannot contain 5 as a factor, for it obviously contains the coefficients of in a degree less by one than that in which jB contains them. It must therefore vanish identically. 97. The results of the preceding article may be confirmed by calculating the actual values of the differential coefficients of R. We know (Art. 67) that R = ^{a)4> {13) 4> (7) &c. But since 4> («) = «««*" + ^w-i^"*"^ + ^^-j ^^ ^*v® ^ = a'; and therefore ' ^ = aV(/3) <^(7)&c. + /S'^(a) ^(7)&c. + &c. If then a satisfies ^, we have <f> (a) = 0, and -r- = of(f> (fi) 4> (7) &c. dR ** In like manner y- = of(f> {13) 4> (7) &c. ; and therefore, as before, dR dR . a dap daq Also, in general, if we multiply together, we find S, S, = "^ ^* (/Q)}' {* (7)1' &c. + 5 (o'/S' + a'^) ^ (7) &c + &c ; and it can easily be seen that the series of terms multiplying jB . d JjL ■«•/» 1 dR dR « 1 • i* t IS -= — 5— . If now we subtract -7 — 7— , the terms not multiplied da/la^ dar da, ' by R will destroy each other if we have p + 2 ~ ^ + 3j ^"^^ there OP TWO EQUATIONS. 93 will then remain dRdR dRdR^^f d*R _ d*R \ da^ doq dor da^ xda^daq da^daj ' T) • •! 1 ^ I X dR dR dR dR . J5y a Bimilar process we can shew that -^ — rt 5 — 57- is divisible by R. but the quotient is not , ,. — -n—jr • aa^bq aatflLOp 98. What has been said is applicable, as we shall presently see, to a system of equations in any number of variables. The following simpler method only applies to a system of two equa- tions. In this case we have seen (Art. 84) that the eliminant can be expressed in the form of the determinant resulting from the elimination of a;*"'\ a;*^, &c. from a system of equations linear in these quantities. When this determinant vanishes, the equations are consistent with each other, and if we leave out any one of them, the remainder will suffice to determine x. Hence if ^,^, /S^,, &c. represent the minors o^ the determinant in question, we have o?"*"^, a;*""", &c. severally proportional to /8jj, ^jg, /8,„ &c., or to i8,j, ;9„, ^^, &c., &c. These values are simpler than those found by the preceding method, since they are a degree lower than the eliminant in the coefficients of each equation ; whereas the values found by differentiating the eliminant are a degree lower than it only in the coefficients of one of the equations. For example, the common value which satisfies the pair of equations aaj* + Jaj + c = 0, aV + J'a; + c' = is by this method found to be — 7 — r = - 7-777 ; whereas by the •^ (ac) (ah) ' ^ preceding method it is given in the less simple form 2c' {cuf) - y {be') _ a' [he') - c' [ab') a' [be') - <f (aV) "" - 2a' (ac') + b' {ab') * All these values are equal in virtue of the relation, which Is supposed to be satisfied, {ac'Y^{ab') {be'), 99. If we substitute in any of the equations used in the last article the values ^ for a;*""*, &c., this equation must be 94 DETERMINATION OF COMMON ROOTS. satisfied when 5 = 0, and therefore the result of substitution must be divisible by B. In other words, if a^^, a^, &c. be the constituents of any of the lines of the determinant of Art. 84, we must have a^, -^ 1- a^ ^ h &c. divisible by B. But if we examine what a^^, &c, are, we see that a,, is the determinant {^J^mJ)i &c., and thus that the function a^,-^ h&c. contains the b coefficients in a degree one higher than B^ while its weight exceeds that of B by n — r+l. Consequently the remaining factor must be h^,r+\ multiplied by a numerical coefficient. To determine this coefficient, we suppose all the terms of -^ to vanish except J_^i. Now it follows at once from the method of elimination by symmetric functions, that if '^ consist of factors F, Wj &c., the eliminant of ^ and -^ is the product of the eliminants of ^, F; ^, TF; &c. For if F be (a; — a) {x-'/3)j &c., and TF be (a?- a')(a: — ^0> &c*? ^^^ eliminant of <f> and F is ^ (a) 4> {/3) &c., that of 4> and TF is ^ (a') {^) &c., and the product of all these is the eliminant of ^ and '^. Again, if ^ reduce to the single term b^x'i^j since the elimi- nant of (f) and X is a^ and of (f) and y is a^, the eliminant of (f) and -^ will be h^a^a^^. The only one then of the series of terms ^ , &c., which will not vanish when all the coefficients ^^'^ dB of ^, except &J,, are made to vanish, will be -i— , and this will be oJ)^*a*~\/» But in the case we are considering, It will be found that the term by which -r- .is multiplied will be b^a^j and hence that in general, when a = w — r + 1, dB dB p , ^N T»T «n T— +a^^-— + &c. = (n-r + l)i?&^,. Ex. In order to make what has been said more intelligible, Iwe repeat the proof for the particular case of the two cubics a^x^ + a^ + a^x + a^ h^ + h^ + h^ + h^ then we have the system of equations (Art. 84) (ogftj) »« + (ff^J,) X + (fla^o) = 0, (a/,) »2 + {(ajJo) + (0361)} X + (agio) = 0, Mo)»*+ («A) a;+(oA) = 0. FOR ANY NUMBER OP VARIABLES. 95 Bubstitnting then, suppose in the second equation, the following quantity must be divisible by ^i But, considering the order and weight of the function in question, it is seen at once that the remaining factor must be b^ multiplied by a numerical coefScient. To deter- mine that coefficient, let &«, b^ b^ all vanish, then the quantity we are discussing Oi ;r-+f*o -rr] • ^^^ ^i ^^ ^^ same supposition, reduces to VVo* i and therefore the function we are calculating at most differs in sign from 2^2^* 100. There is no diflSculty in . applying the method of Arts. 96, 97 to the case of any number of variables. For greater clearness we confine ourselves to three variables, but the same proof applies word for word to any number. Let there be three equations = 0, -^ = 0, x~^» where ^ = ^mo o^'" + '**+^» 5 ixi'y^z'^ + &c.^ and let the values ar'yV satisfy all the equations; then they will still satisfy them if in 4> we alter a^,^„ a^^s^y into a^,,,+ ^^^,^„ aa,^,^ + ^,,i3,7, &c, provided only that A^^^x^ -f &c. + A^^^^^x'^^yf^z"* + &c. = 0. But, as in Art. 96, the equation must also be satisfied and comparing these two equations, we see that the value of each term x'^^yf^z'^ must be proportional to the differential of the eliminant with respect to the coefficient which multiplies it. We obtain the values of x\ y', z\ by taking the ratios of the differentials of jB with respect to the coefficients of any terms which are in the ratio of a;, y, 0. And this may be verified as in Art. 97. For let the common roots of ^, X'i substituted in 0, give results 0', 0", &c. Then B = 0>'>'" &c. And ^^ ^ a: V'^'^f T' &c. + aj'V^«^''f f " &c. + &c. ; and if we suppose ^' to vanish, the value of this differential coefficient reduces to its first term, and it is seen, as before, that the differential with regard to each coefficient is proportional to the term which that coefficient multiplies. The same corol- laries may be drawn as in Art. 96. 101. And generally, in like manner, if the coefficients of be functions of any quantities a, i, c, &c., which do 96 DETERMINATION OP COMMON BOOTS. not enter into -^j Xi *^ ^^ proved by the same method that dS dR d<f> dd> , • ^i t •••/*• .1 ir'dh''7r'^^ where m the latter differentials aj, y, z are supposed to have the values a;', y, «', which satisfy all the equations. For either, as in Art. 97, we have when ^^ = 0, "^ " ^ '^''*'' *^" ^ " ^ *"*'" *^- ' ^^' *^^^°' ^ ^° -^' ^^' if a, i, c be varied, so that the same system of values continues to satisfy ^, we have while, because in this case the eliminant of the transformed ^ and of the other equations continues to vanish, we have dJR ^ dJib m,y dR m, o -J- oa-\--TLOO'V 3-oc + &c. = 0, da do dc ' and these two equations must be identical. 102. The formulae become more complicated if we take the differentials of the eliminant with respect to quantities a, &, &c. which enter into all the equations. As before, if we give these quantities variations, consistent with the supposition that the eliminant still vanishes, we have dR . dR ^m dR ^ f% -3- oa + -jT 00 + -J- oc 4 &c. =s 0. aa do ac Now, in the former case, where a, 5, c, &c. only entered into one of the equations, a change in these quantities produced no change in the value of the common roots, since the coefficients remained constant in the other equations, whose system of common roots was therefore fixed and determinate. But this will now no longer be the case, and the common roots of the transformed equations may be different from those of the original system. Let the new system of common roots be 0?' + Saj', y' + Sy', «' + iz\ &c., then the variations are connected by the relations ^Sa + ^86 + &c. + g-5a5' + ^S/ + &c.=0, TWO COMMON SYSTEMS. 97 If there are h such equations, there will be Aj — 1 independent variables ;* we may, therefore, between these h relations elimi- nate the ^ — 1 variations hx\ hy\ &c., and so arrive at a relation between the variations Sa, SJ, &c. only ; the coefficients of which must be severally proportional to -r- , -^ , &c. Ex. 1. Let there be two eqnatioDs and one variable. The final relation then is and the several coefficients are proportional ^o ^ , -^r-, dec. If the equation had been given in the homogeneous form, we might have taken x as constant, and sub- Btituted "^J •^'^^^» ^"^ precedmg formula. This makes no change, because it was proved, Art 89, that the common root satisfies the Jacobiani or makes d<f> dy^f d<f> dxj/ dx dy" dy dx' Ex. 2. If there are three equations, the coefficient of da is d4> d^ dx da ^ da ^ da *Pif ^11 Xi <t>9» ^29 Xa where 0|, 02 denote the differential coefficients of 4> with respect to x and y, Ac. 103. If a system of equations is satisfied by two common systems of values, not only will the eliminant B vanish, but also the differential of R with respect to every coefficient in either equation. For evidently the values of the differentials, given Art. 97, all vanish if both <f> (a) and <f> (^3) = 0, or, in Airt. 100, if ^', 4>' both = 0. In this case the actual values of the two common roots can be expressed by a quadratic equa- tion in terms of the second differentials of R. The following, though for brevity, stated only for the case of two equations, applies word for word in general. We have (see Art. 97) i^=af'/y^(7) ^(8)&c. + /3V^(a) ^(8)&c. + &c., ♦ If the equations had been given as homogeneous functions of h variables, still eince their ratios are all we are concerned with, we may assume any one of the yariables ^ to be the same in all the equations, and may suppose it^ = 0. O 98 DISCRIMINANTS. r which, when ^ (a), ^ {0) = 0, reduces to the single term o^fi'<f> (7) <f> (S) &c. In like manner, in the same case, d^=('^^-^''*^'f'iy)'f>^^)^''-^ i ^ = a*/3'^ (7) ^ (8) &c If then we solve the quadratic in X : /t, ^d*B ^ d*R ^d*B_ da/ da^dug da/ ' the roots will give the ratios 0? : a*, fi^ : ^. If the equations have three common systems of values, all the second differentials of R vanish, and the common roots are. found by proceedmg to the third differential coefficients and solving a cubic equation. LESSON XL DISCRIMINANTS. 104. Before entering on the subject of discriminants, we shall explain some terms and symbols which we shall frequently find it convenient to employ. In ordinary algebra we are wholly concerned with equations^ the object usually being to find the values of x which will make a given function =0. In what follows we have little to do with equations, the most frequent subject of investigation being that on which we enter in the next Lesson : namely, the discovery of those properties of a function which are unaltered by linear transformations. It is convenient| then, to have a word to denote the function itself, without being obliged to speak of the equation got by putting the function = : a word, for example, to denote aa;* + 5a;y + cy' without being obliged to speak of the quadratic equation <3:£c' + 5a?y + cy*=0. We shall, after Prof. Cayley, use the term qiuintic to denote a homogeneous function in general; using the words quadriC| cubic, quartic, quintic, n% to denote quantics of the 2nd, 3rd, 4th, 5th, n*^ degrees. And we distinguish quantics into binary, ternary, quaternary, w-ary, according as they contain 2, 3, 4, THEIB OBDER AND WEIGHT. 99 n variables. Thos, bj a binary cubic, we mean a function Buch as aaj' + fta5*y + Cicy* + rfy' ; by a ternary quadric, such as CO? + 5y* + c«* + ^fyz + ^gzx + 2Aa?y, &c. Professor Cayley uses the abbreviation (a, 5, c, e^X^'} vf ^^ denote the quantic aaj' + 35aj*y + Sexy' + c?y', in which, as is usually most con- venient, the terms are affected with the same numerical coeffi- cients as in the expansion of {x^yf. So the ternary quadric written above would be expressed (a, J, c, /, g^ AJa?, y, zf. When the terms are not thus affected with numerical coeffi- cients, he puts an arrow-head on the parenthesis, writing, for instance (a, 5, c, S7§Xi yf to denote aa^ + ho^y + crjf + d\^. When it is not necessary to mention the coefficients, the quantio of the n** degree is written (a;, y)*, (a;, y, a)*, &c. 105. If a quantic in h variables be differentiated with respect to each of the variables, the eliminant of the k differentials is called the discriminant of the given quantic. If n he the degree of the quantic.^ its discriminant is a homo* geneous function of its coefficients^ and is of the order ^ (n — !)*"*• For the discriminant is the eliminant of k equations of the (h — 1)** degree, and (Art. 78) must contain the coefficients of each of these equations in a degree equal to the product of the degrees of all the rest, that is (n- 1)*"^ And since each of these equations contains the coefficients of the original quantio in the first degree, the discriminant contains them in the A (n — 1)*^. degree. Thus, then, the discriminant of a binary quantic is of the degree 2 (n — 1) ; of a ternary, is of the degree 3(n-l)",&c 106. Jfin the original quantic every coefficient multiplying the first power of one of the variahles x he affected with a suffix 1, every term multiplying the second power by a suffix 2, and so on ; then the sum of the suffixes in each term of the discriminant is constant and =n(n— 1)*^. It was proved (Art. 78) that if every coefficient in a system of equations were affected with a suffix corresponding to the power of x which it multiplies, then the sum of the suffixes in every term of their eliminant would be equal to the product of the degrees of those equations, viz., =: mnp &c. Now suppose, that in the first of these equations 100 DISCRIMINANTS. the suffix to the coefficient o{x^^ instead of being 0, was Z; that of 05* was Z+ 1, and so on ; it is evident that the effect would be to increase the sum of the suffixes by I for every coefficient of the first equation which enters into the elirainant; and since (Art. 78) every term contains np &o, coefficients of the first equa- tion, the total sum of suffixes is mnp &c. 4 Inp&c. = (wi + Z) np &c. Now, in the present example, it is evident that every coefficient in the ft— 1 differentials U^^ C^, &c.,* multiplies the same power of X ai^ it did in the original quantic U. But in the remaining differential, i7„ every coefficient multiplies a power of x one less than in 27, and the coefficient multiplying any term of in this differential will be marked w^ith the suffix Z+l, since it arose from differentiating a term a;'^^ in the original quantic. It follows, then, that the sum of suffixes in the discriminant must = (w - 1)*+ {n - 1)*"* = n (w - 1)*^'. We shall briefly express the results of this and of the last article by saying that the order of the discriminant is k{n — 1)*^ ; and its weighty n (w — 1 j*'*. Thus for a binary quantic the weight of the discriminant is n (n - 1). 107. If a binary quantic contain a square factor, then, as is well known, the discriminant vanishes identically. For the two differentials must each contain that factor in the first degree, and therefore, since they have a common factor, their eliminant vanishes. In like manner, if a ternary quantic be of the form X*</» -f Xr^ + Y\^ where X=^ax-^by-{- cz^ r= a'x + Vy + c'a, then the discriminant must vanish, since every term in any of the differentials must contain either X or Y^ and therefore the differentials have common the system of roots derived from the equations X=0, F=0. In like manner, the discriminant of a quaternary quantic vanishes, if the quantic can be expressed as a function in the second degree of X, Y^ Z^ these being any linear functions of the variables.t We shall call those values * We write, as before, U^j U^, U^j &c, to denote the difEerential coefficients of U with respect to x, y, Zf &g. t In other words, the vanishing of the discriminant of an algebraical equation expresses the condition that the equation shall have equal roots ; and the vanishing pf the discriminant of the equation of a curve or surface expresses the condition that the curve or surface shall have a double point. IN TERMS OF THE BOOTS. 101 which make all the differentials vanish, the singular roots of the quantic. 108. We shall now discoss the properties of the discriminant of the binary quantic 17= a^x''^- na^c^y + \n {n-l)agaj""*y'+&c. The eliminant of U and U^ is a^ times the discriminant^ and the eliminant of U and U^ is a^ times the discriminant,* •. For since nU^xU^-^yU^^ the result of substituting in nUanj root of C^ is y'£^'; and when all the results of substitution are multiplied together, the product will be y'y"y[" &c. (which is = a^, see Art. 61), multiplied by the product of the results of substituting the same roots in £^, which is the discriminant. 109. To exfpress the discriminant in terms of the values ^1^1 J ^8^2* ^ J which make the quantic vanish. Let Z7= [xy^ - yx,) {xy^ - yx^) {xy^ - yx^ &c. (see Art. 61) .; then ^i=yi(a3^8-y^«)(^y8-y^>) &c.+y,(aryj-2^x,)(icy3-2^a?3) &c.4&c. ; and the result of the substitution in 17^ of any root x^y^ of U is y, (ojjy,— y^icj (iCj^j — ^jiCj) &c. Similarly, the result of sub- stituting xj/^ is y^ [xj/^ - a;,yj {xj/^-y^x^ &c. If, then, all the results of substitution are multiplied together, the product is ± ^1^82^8 &c. {xjf^ - y,x^^ [x^y, - y.x^f [xj/^ - y^x^j" &c. This, then, is the eliminant of U and £^, and if we divide it by a^, which is =y,yj.y8^^*j ^® ^9\\ have the discriminant = (^i^a "■ yj^a)' (^1^8 "" ^1^8)* ^^* ^^ ^® make in it all the y's = 1, we get the theorem in the well-known form that the discriminant is equal to the product of the squares of all the differences of any two roots of the equation. We shall, for simplicity, refer to the theorem in the latter form. 110. The discriminant of the product of two quxintics is equal to the product of their discriminants multiplied by the square of. their eliminant. For the product of the squares of differences of all the roots evidently consists of the product of the squares of differences of two roots both belonging to the same quantic, * We do not take account of mere numerical factors. 102 DISCRIMINANTS. moltiplied by the square of the prodact of all differences between a root of one and a root of the other, and this latter product is . the eliminant (Art 68). As a particular case of this, the dis- criminant of (a; — a) <f> [x) is the discriminant of ^ {x) multiplied by the square of ^ (a). For if )8, 7, &c. be the roots of ^ {x)j then (a-i8)'(a — 7)*{/8 — 7)* &c. is equal to the square of (a — i8) (a — 7) &c. which is ^(a), multiplied by the product of the squares of all differences not containing a. 111. ITie diacrtminant of (a,, a^...a^^,, ^m!K^) hT *^ ^f ^^^ form ajif) + «*„_,^, where -^ is the discriminant of the equation of the [n - 1)** degree (a^, a,...a^.^, ^n-i5C^i VT^* ^^^ w® evidently get the same result, whether we put any term a^ = in the discriminant, or first put a^ = in the quantic, and then form the discriminant. But if we make a^ = in the quantic, we get X multiplied by the (n— 1)*" written above, and (Art. 110) its discriminant will then be the discriminant of that [n — lY multiplied by the square of the result of making in it a; = 0; that is, by the square of a^_^. In like manner we see that the discriminant is of the form ajl> + a^y^.* 112. The discriminant being a function of the determi- nants x^y^ - a;,^j, &c must satisfy the two differential equations (Art. 62), dA ^ dA ^ dA I, '0 ^^i ^^t or, if the original equation had been written with binomial coefficients, rfA , X d^ o dA e?A » * This theorem was first published by Joachimsthal ; I had, however, previously been led by simple geometrical considerations to the foUo^ring theorem in which it is included. If a^ contain a factor e, and if a^ contain 2^ as a factor, the discriminant will be divisible by «'. If a^ contain 2; as a factor, if a, contain a*, and a^ contain «•, the discriminant will in general be divisible by sfi. In like manner, if a, contaia z ; O31 s* ; On £* ; and a^ e^, the discriminant will be divisible by z^^, &c. DETERMINATION OP EQUAL ROOTS. 103 Ex. To form the discriminant of (og, Oi, a„ ...]j^«) ^)", which we suppose arranged according to the powers of a^. We know (Art. Ill) that the absolute term ia Oi^Df where D is the discriminant of (oi, a^ ...^jfrr, y)*"^. The discriminant then ia OiW + a^4» + a^^lr + Ac. ; operating on this with «i t— + 2a, t- + 3a, t— ■ + 4a, we may equate separately to zero the coefficient of each power of cl^ Thus, then, the part independent of a^ is a^ti^ + AayoJD + a^ Ua^ ^ + 8a, ^ + 4c.) D ; or, remembering that (^ ^ + 2a» ^ + Ac*) -D = 0, we have <^ = -4a,i) + a,(a,^ + 2a,^^ + 4c.)2>, and the discriminant is In like manner, from the coefficient of a| we can determine ^, but the result does not seem simple enough to be worth writing down« (a,« - 4aoaj) D + a^a^ f*^ ^ + ^« ^ + ^ ^ + ^*^ + *^ 113. K the discriminant of a binary quantic vanishes, the quantic has eqaal roots, and the actual valaes of these roots can be found by a process similar to that employed in Lesson x. Let J7=a^a5*-f a,aj'*"* + agaj*'* + &c. be a quantic whose discrimi- nant vanishes, and having therefore a square factor (a? — a)\ Then evidently F, where will also be divisible by a- a, provided that -4^, -4^, &c. be any quantities satisfying the condition Aji^ + AfiT'^ + AjT" + &c. = 0. In this case then we shall have U^-W divisible by a; — a. Let it =(aj — a) {(a;V a) ^(a?) + X^(a?)}. It follows then, from Art. Ill, that the discriminant of 27+ XF is the discriminant of the quantity within the brackets, multiplied by the square of the result of substituting a for x inside the brackets. But this result is X^(a). We have proved then that in the case supposed, the discriminant of Z7+ XF is divisible by X*. But since Z7+XF is derived from U by altering a^ into flr^ + X-4p, &c., the discriminant of Z7-f XF is derived from the discriminant of Uhy 2^ like substitution, and is therefore 104 DISCRIMINANTS By hypothesis A = 0. Bat the discriminant will not be divisible by \* unless the coeflScient of X vanish. Now the relation thus obtained between A^^ ^„ &c. must be identical with the relation A^a* + -4,a**~* + &c. = 0, which we have already seen is the only relation that need be satisfied by A^^ A^^ &c. in order that the discriminant of J7+XFmay be divisible by X*. We must have therefore the quantities a*, a"'^, a**"", &c. respectively propor- tional to -T- , -T- ) -T— ) &c. Dividing any one of these terms by that consecutive to it, we get an expression for a. We may state this result : When the discriminant vanishes^ the several dif- ferential coefficients of the discriminant with respect to a^,, a„ &c. are proportional to the differential coefficients of the quantic with reject to the same quantities. 114. This result may be confirmed by forming the actual values of these differential coefficients in terms of the roots, which may be done by solving from the n equations <?A _ <?A c?a, (iA da^ « doL da^ doL da^ da We know the expressions for A, a,, a^, &c. in terms of the roots, and therefore from these n equations can find the n quantities sought. The result will be found to be ^ = 2(/3-7r(7-Sr(S-/8)" X {(a-/S) {a-7)+ («-y8) (a- 8) + (a-7) (a- S)}, where the product of the squares of all the differences, not con- taining a, is multiplied by the sum of the products (n — 2 taken together) of the differences which contain oe, ^=Sa(^-7)'(y-8r(«-i8r{(a-/3)(a-7) + &c}, da. = 2a' (/8 - if (7 - «)* (S - /S)' {(a - /S) (a - 7) + «&c.}, &c, fl-2 and the supposition a = /8 reduces these sums to quantities which are in the ratio 1, a, a^, &c. As in Art. 96, it follows from the FOB ANY NUMBEB OP VARIABLES. 106 theorem of tbe last article that -^ — = 5 — j- is divisible bv cuip oa, da, da, ^ A when p + j = r + «. If more than two of the roots are equal to each other, all these differentials vanish identically, and we find the eqnal roots by proceeding to second differentials of the discriminant. 115. We know, from Art. 98, that instead of the functions in the last two articles, which are of an order in the coefficients only one lower than the discriminant, we may substitute func* tions of an order two lower, and possessing the same property^ viz. that they vanish when more than two roots are equal, and that if two roots are equal (a = )8) they are to each other in the ratios 1, a, a*, &c. K we proceed by Bezout's method of elimination (Art. S4) to eliminate between the two different tials £^, t^, the resulting equations of the (n — 2)* degree, when expressed in terms of the roots, are 2 (a — ^8)* {x — 7) (a? — 8) = 0, 2j, {a--/3)»(aj-7)(aj-S)=0, 2yja-/3)«&c. = 0, where J„j.,&c. are the sum, sum of products in pairs, &c. of all the roots except a and ^3.* The discriminant is then, by Bezout's method, expressed as a determinant, whose constituents are 2 (a-/S)«, 2j, (a-/3r, 2?, (a-./S)«, &c,, 2j, (a - P)\ 2j.« (a - ^)\ 2j,j. (a - /S)', &c., 2?.(a-/3)«, 2j,j,(a-/3r, 2?.« (a - /J^, &c., &c. And when the given equation has two roots equal, the first minors of this determinant will, by Art. 98, be in the ratio 1, a, a', &c. A somewhat simpler series, possessing the same property, is 2 (/3 - 7^ (7 - «)' (« - /3)», 2a 09-7)' (7-8/ (8-/3/, 2a*(/S-7)'&c. 116. The following proof of the theorem of Art. 113 is applicable to the case of a quantic in any number of variables. For simplicity, we confine ourselves to the case of two inde- pendent variables, the method, which is that of Art. 102, being ♦ The first of these functions of degree « — 2 is one of the series to which we are led by Sturm's process. With regard to the extension of Sturm's theorem, see Sjlvester's memoir in the Philosophical Transactiont for 1853. P r 106 DISCRIMINANT8. equally applicable in general. Let the coefficients in £7 be functions of any quantities a, b, &c., and let variations be given to these quantities consistent with the supposition that the dis- criminant still vanishes, and therefore such that -T- 5a + -jr S5 -f &c. = 0. aa do Now if the effect of this change in a, 5, &c.y is to alter the singular roots from x^ y into x-\'hx^ y + hy^ since these new values satisfy £^, D^, £^, &c., we must have -j-^ oa + -j7-' 0* + &c. + -7 * oaj + -r-^ oy = 0; aa ab dx ay Multiply these equations by a;, y, z respectively and add; then since nU^xU^^-yU^-^ zU^^ the coefficient of ia will be n -y ; and since —^-^ = -7—' , —7-^ = -^ , the coefficient of 8» aa dx ay ^ dx dz ^ will be (n — 1) C^, which will vanish, since U^ is satisfied for the singular roots. We get therefore dU^ dU^^^^ ^ -7— oa + -77- 06 + &c. = 0, da do ^ and therefore the differentials of A with respect to a, J, &c. are proportional to the differentials of U with respect to the same quantities, it being understood that the a;, y, z which occur ia the latter differentials are the singular roots. 117. The theorem proved for binary quantics (Art. Ill) may be extended to quantics in general. Let a be the coefficient of the highest power of any of the variables, J, c, c?, &c., those' of the terms involving the next highest power, then the dis- criminant is of the form ad + (^, X'i '^j &c.3[6, c, cZ, &c.)'. Thus, for a ternary quantic, to which for greater simplicity we confine ourselves, if a be the coefficient of «* ; J, c those of FOR ANT NUMBER OF VARIABLES. 107 «*"^a;, «**~*y ; then if in the discriminant we make a = 0, the re- maining part will be of the form 6*^ + Jc^ + c';^. To prove this : first, let U be any quantic whose discriminant vanishes, V any other reduced to zero by the singular roots of Z7, then I say that the discriminant of CT-f \V will be divisible by X*. For, let Z7=a«*4-fe""'a: + &c., 7= ^«" + ^^""'a? + &c., then the coefficient of X in the discriminant of Z7+XF will be A^ — hjB-^4-&c., and (Art. 116) -r- ^ &c. are proportional to «*, «"'*a:, &c. The coefficient of X is therefore proportional to the result of substituting the singular roots in F, and there- fore vanishes. Now, in the case we are considering, the supposition of <2=i0y i = 0, c = must make the discriminant vanish, since then all the differentials vanish for the singular roots a; = 0, y = 0. Any other quantic V will vanish for the same values, provided only ufl «= 0. The general form of the discriminant then must be such that if we substitate for 2>, i 4 "KB ; for c, C + XC7, &c., and then make a, 2>, c = 0, the result must be divisible by X*; or, in other words, if we put for J, XJ5; for c, XC7, &c., and then make a = 0, the result is divisible by X', which was the thing to be proved. 118. Concerning discriminants in general, it only remains to notice that the discriminant of a quadratic function in any number of variables is immediately expressed as a symmetrical determinant. And, conversely, from any symmetrical deter- minant, we may form a quadratic fanction which shall have that determinant for its discriminant. The simplest notation for the coefficients of a quadratic function is to use a double suffix, writing the coefficients of a?*, y*^ &c., a^,, a^^ a^^ &c., and those of xy^ xz^ &c., a^^, a,g ; a^^ and a^^ being identical in this notation. The discriminant Is then obviously the sym- metrical determinant «ji» ««? «i»? &c- «8i) «w? ««8) &C' «8i) «»a) «8B) &c- &c. ( 108 ) LESSON XIL UNEAB TBANSFOBMATIOI^S, 119. Invariants. The discriminant of a binary qnantio being a function of the differences of the roots is evidently unaltered when all the roots are increased or diminished by the same quantity. |9^ow the substitution of a; + X for a; is a particular case of the general linear transformation^ where, in a homogeneous function, we substitute for each variable a linear function of the variables; as for example, in the case of a binary quantic where we substitute for x^ Xx-^fit/y and for ffj Wx-Vfi^y. It will illustrate the nature of the enquiries io which we shall presently engage if we examine the effect of this substitution on the discriminant of the binary quadratic, aia?'\'2bxy-\-oy^. When the variables are transformed, it be« comes a (Xo? + /lAy)" + 25 (Xaj + Aty) ( Voj + /y) + (X'o? + /y)" ; aod if we call the transformed equation aV+2i'ajy + {^y, we have a' = aX* + 25\X' + cX'*, o' = a/*' + 2 J/ia/ + o/ia'*, 6' = aX/A + J(X/ + X» + cXV. It cau now be verified without difficulty that that is to say, the discriminant of the transformed quadratic is equal to the discriminant of the given quadratic multiplied by the square of the determinant X/^'-X'/^, which is called the modulus of transformation* 120. Now, a corresponding theorem is true for the discrimi- nant of any binary quantic« We can see h priori that this must be the case, for if a given quantic has a square factor, it will have a square factor still when it is transformed; so that whenever the discriminant of the given quantic vanishes, INVARIANCE. 109 that of the transformed must necessarily vanish too. The one must therefore contain the other as a factor. The theorem however can be formally proved as follows: Let the original quantic be {xy^^yx^{xy^-yx^&ii,^ then (Art. 109J the dis- criminant is [xjf^ - y^x^'' (aj,y, - y^x^^ &c. Now the linear factor [xy^ — yx^ of the given quantic be- comes by transformation y, (XX+/tF) — a?, (X'X+/l6'F), and if we write this in the form Y^X-X^Y^ we shall have Fi = Xy, — X'oj,, Xj » — /liy, + /Lt'oj,. If then the transformed quantic be written as the product of the linear factors (F^X- X^Y) [Y^X- -^8^) &C'j ^® \i9>yQ expressions, as abovoi for r„ X^ ; F,, X„ &c., in terms of y^, x^ ; y„ a?,, &c. We can then, without difficulty, verify that It foUows immediately that (r,Z,-F,Z/(r,X3-F3ZJ»&c. is equal to {n^x^ — xj/^^iy^x^'-'y^x^'^&Q. multiplied by a power of X/jf — X^fi equal to the number of factors in the expression for the discriminant in terms of the roots. A corresponding theorem is true for the discriminant of a quantic in any number of variables. What I have called Modern Algebra may be said to have taken its origin from a paper in the Cambridge Mathematical Journal for Nov. 1841, where Dr. Boole established the prin- ciples just stated and made some important applications of them. Subsequently Prof. Cayley proposed to himself the problem to determine h priori what functions of the coefficients of a given equation possess this property of invariance\ that when the equation is linearly transformed, the same func- tion of the new coefficients is equal to the given function ^lultiplied by a quantity independent of the coefficients. The result of his investigations was to discover that this property of invariance is not peculiar to discriminants and to bring to light other important functions (some of them involving the variables as well as the coefficients) whose relations to the given equation are unaffected by linear transformation. In explaining this theory, even where, for brevity, we write only three variables, the reader is to understand that the 110 LINEAR TRANSFORMATIONS. processes are all applicable in exactly the same way to any number of variables. 121. We suppose then that the variables in any homo- geneous quantic in k variables are transformed by the substitution X = \x + A*, r+ v,Z+ &c., y = \X + /*, r+ r,Z+ &c., and we denote by A the modulus of transformation ; namely, the determinant, whose constituents are the coefficients of transformation, \^ /*,, v^, &c., \^ /jl^^ v^, &c., &c. Now it is evidently not possible in general so to choose the coefficients X„ /a,, &c., that a certain given function ax* + &c. shall assume, by transformation, another given form a'X" + &c. In fact, if we make the substitution in aa5"4-&c., and then equate coefficients, we obtain, as in Art. 119, a series of equa- tions a' = a\^ + &c., the number of which will be equal to the number of terms in the general function of the n** degree in k variables. And to satisfy these equations we have only at our disposal the k* constants \„ X^, &c., a number which will in general be less than the number of equations to be satisfied.* It follows then that when a function aa;** + &c. is capable of being transformed into a'X" + &c., there will be relations con- necting the coefficients a, i, &c., a', b\ &c. In fact, we have only to eliminate the k^ constants from any i* + 1 of the equations a' = a\* + &c., and we obtain a series of relations connecting a, a', &c., which will be equivalent to as many independent relations as the excess over k* of the number of equations. Thus, in the case of a binary quantic, the number * The number of terms in the general equation of the n*^ degree homogeneoufl - T '■,.■, . (« + 1) (« + 2)...(n + k — 1) , .^ . . i.1- i. XT. 1 in Jb variables is -^ ^-^-^ — /, ... -, and it is easy to see that the only cases where this number is not greater than k^ are, first, when n - 2, when it becomes ^k {k + 1), a number necessarily less than k^y k being an integer ; and secondly, the case k = 2f n = 3, when both numbers have the same value 4. That is to say, the only cases where a given function can be made by transformation to assume any assigned form are, first, the case of a quadratic function in any number of TftriableB; and secondly, the case of a cubic function homogeneous in two variables. ABSOLUTE INVARIANTS. Ill of terms in a homogeneous function of the vl^ degree is n + !• If then, in any quantic aaj* + &c., we substitute for a;, \X^r a*, Y^ and for y, \X+/l6jF, and if we then equate coefficients with o'^'^+iifcc., we have n + 1 equations connecting a, a', X„ &c., from which, if we eliminate the four quantities X„ X^, ft,, ft,, we get a system equivalent to n — 3 independent relations between a, 2>, a\ b\ &c. It will appear in the sequel that these relations can be thrown into the form <f> (a, 5, &c.) = <f> (a', J', &c.) ; or, in other words, that there are functions of the coefficients a, 5, &c. which are equal to the same functions of the trans* formed coefficients. The process indicated in this article Is not that which we shall actually employ in order to find such functions, but it gives an h priori explanation of the existence of such functions, and it shows what number of such functions, independent of each other, we may expect to find. 122. Any function of the coefficients of a quantic is called an invariant^ if, when the quantic is linearly transformed, the same function of the new coefficients is equal to the old function multiplied by some power of the modulus of transformation; that is to say, when we have j> (a', 5', c', &c.) = ^^<^ (a, i, c, &c.). Such a function is said to be an absolute invariant when^ = 0; that is to say, when the function is absolutely unaltered by transformation even though A be not =1. If a quantic have two ordinary invariants, it is easy to deduce from them an absolute invariant. For if it have an invariant 4>, which when transformed becomes multiplied by A'', and another '^, which when transformed becomes niultiplied by A*, then evidently the 2** power of ^ divided by the p*^ power of -^ will be a function which will be absolutely unchanged by transformation. It follows, from what has been just said, that a binary quadratic or cubic can have no invariant but the discrimi- nant, which we saw (Art. 120) is an invariant. For if there were a second, we could from the two deduce a relation ^ (a, 6, &c.) = ^ (a', &', &c.). But we see from Art. 121 that there can be no relation connecting a, 5, &c. with a\ h\ &c., 112 LINEAR TRANSFORMATIONS. since, with the help of the fonr constants \,, &c. at our dis** posal, we can transform a given quadratic or cubic, so that the coefficients of the transformed equation may have any values we please. In the same manner we see that a quantic of the second order in any number of variables can have no invariant but the discriminant. On the other hand, suppose we take the binary quartic aa?* + 46a;*y + 6cajy + 4diry' + ey*, and that the coefficients become by linear transformation a', y, &c., it will be found that we have two invariant functions both distinct from the discriminant ; in fact, we have the two equations aV - ^Vd' + 3c'* = A* {ae - 4JJ + 3c»), a'cV+2JVeZ'-a'rf'«-e'&'*-c'»=A'(ace + 25ci-a(r-^«-c*), and from these two we deduce the absolute bvariant (aV-4j'd' + 3c'7 (ae-.4M+3cy In this case the invariance of the discriminant may be deduced as a consequence of the preceding equations, for the discri- minant is (ae - 43)d + 3c*)' - 27 {ace + 2 Jcrf - oeZ" - ci* - c^«, and consequently the discriminant of the transformed equation is equal to that of the original multiplied by A". 123. In the same manner as we have invariants of a single quantic we may have invariants of a system of quantics. Let there be any number of simultaneous equations oa?" + &c. = 0, do? + &c. = 0, &c., and if when the variables in all are trans- formed by the same substitution, these become AX^ + &c. = 0, ^'X* + &c. = 0, &c., then any function of the coefficients is an invariant if the same function of the new coefficients is equal to the old function multiplied by a power of the modulus of transformation ; that is to say, if ^ (-4, 5, &c., A\ 5',&c.,J-4",&c.) = £^^^[a^h^&Q.^a\l\ &c., a", &C.), The simplest example of such invariants is the case of a system of linear equations. The determinant of such a system INVARIANTS OF A STSTEM. 113 18 an invariant of the system. This is evident at once A*om the definition of an invariant and from the form in which the fundamental theorem for the multiplication of determinants has been stated in Art. 23. If we are given an invariant of a single quantic, we can derive from it a series of invariants of systems of quantics of the same degree. ' In order to make the spirit of the method more clear, we illustrate it in the first instance by a simple example. We have seen (Art. 119) that oc — J* is an in- variant of the quadratic ax^ + 2bx}/ + cy*j and we shall now thence derive an invariant of a system of two quadratics. Suppose that by a linear transformation ax^ + 2bxy + cy* becomes AX* + 2BXY-^CY\ and aV + 2ya?y + cy becomes ^'X'+25'Xr+CT*; then evidently, by the same transfor- mation {k being any constant), (a + ka') a? + 2{b + kb')xy\-{c-^ hf) y" will become (A+X;^')X«+2(5+*5')-^r+{a+AC') Y\ Forming then the invariant of the last quadratic, we have (Art 119) (-4 + A^'){C'+*t7')-(^+*^')-A*{(a+Aa')(c+*cO-(i+A:JO'l- But since k is arbitrary, the coefficients of the respective powers of k must be equal on both sides of the equation ; and therefore we have not only, as we knew before, (^C-.5») = A'(ac-i'), (.l'C'-5^) = A'(«V-0, but also AG' + CA' - 2BB' = A« [ac' -^ca'-- 2 JJ'), an equation which may also be directly verified by the values of -4, 5, &c. given Art. 119. We see then that oc' + ca'- 2J6' is an invariant. By exactly the same method, if we have any invariant of a qnantic ox" + &c., and if we want to form invariants of the system oa?" + &c., a'a;" + &c., we have only to substitute in the given invariant for each coefficient a^a-\- kd^ for J, J + kh\ &c., and the coefficient of each power of k in the result will be an invariant. Writing down, by Taylor's theorem, the result of Q i 114 LINEAR TRANSFORMATIONS. substltating a + Tea' for a, &c., the theorem to which \re have been led may be stated thus: If we have any invariant of a qaantic aa;" + &c., and if we perform on it the operation H' fL a' -T- + i' ;^ + &c., we get an invariant of the system of two qaantics oa?" + &c., cl^ + &c. We may repeat the same opera- tion and thus get another invariant of the system, or we may H fL operate with a'' -7- + J" ^ + &c., and thus get an invariant of a system of three quantics, and so on. This latter process gives us the invariants which we should find by substituting for a^a-^ ho! + Za", &c., and taking the coefficients of the pro- ducts of every power of Ic and h In the same manner we get invariants of a system of any number of quantics. 124. Covariants, A covariant Is a function involving not only the coefficients of a quantic, but also the variables, and such that when the quantic is linearly transformed, the same function of the new variables and coefficients shall be equal to the old function multiplied by some power of the modulus of transformation ; that is to say, if ax* + &c. when transformed becomes AX"" + &c., a function ^ will be a covariant* if it is such that ^{A^ By &c., Xj Yy &c.) = A^<f) (a, J, &c,, a?, y, &c.). Every invariant of a covariant is an invariant of the original quantic. This follows at once from the definitions. Let the quantic be aaj" -f &c., and the covariant «'«"* + &c. which are supposed to become by transformation JX* + &c., -4'JE"* + &c. Now an invariant of the covariant is a function of its coefficients such that ^ (J.', £', &c.) = ^"0 {a\ V, &c.). * In the geometry of curves and surfaces, all transformations of cooidinafceB are effected by linear substitution. An inyariant of a ternary or quaternary quantic ia a function of the coefficients, whose vanishing expresses some property of the curve I or surface independent of the axes to which it is referred, as, for instance, that the curve or surface should have a double point. A covariant will denote another curve or surface, the locus of a point whose relation to the given curve is independent of the choice of axes. Hence the geometrical importance of the theory of inyuiaiitii imd oovariantfl. EMANANTS ARE COVARIANTS. 115 But A\ B\ &c. by definition can only differ by a power of the modulus from being the same functions of A^ Bj &c. that a\ V^ &c. are of a, h^ &c. Hence when the functions are both expressed In terms of the coefficients of the original quantic and its transformed, we have ^ (-4, -B, &c,) = A'^ (a, J, &c.), or the function is an invariant. Similarly, a covarlant of a covariant is a covariant of the original quantic. 125. We shall in this and the next article establish prin- ciples which lead to an important series of oovariants. If in any quantic u we substitute x + hx! for a;, y + hyf for y, &c., where x'y'z' are cogredient to ocyz^ then the coeffi- cients of the several powers of h^ which are all of the form a?'-r- +y' Tr- + &c. J w, have been called the first, second, third, &c. emanants* of the quantic. Now eaaJt of these emananta is a covariant of the quantic. We evidently get the same result whether in any quantic we write x + kx^ for a;, &c., and then transform x^ afj &c. by linear substitutions, or whether we make the substitutions first and then write X-^kX^ for JT, &c. For plainly \X+fi,Y+ y,Z^ k {\X' + M. r + y,Z') r.\{X+kX')-^fi,{Y'^kY') + y,{Z+kZ'). If then u becomes by transformation Z7, we have proved that the result of writing x + kx^ for a;, &c. in u, must be the same as the result of writing X+kX^ for X, &c. in Z7, and since k is indeterminate, the coefficients of k must be equal on both sides of the equation ; or , du , du p ^, dU ^rf dU . « « ^ « ^ a! 3- + /-r + &c.= Z' rr^ + r'^^-F &c., &c. Q.E.D. dx ^ dy dX di ' 126. If we regard any emanant as a function of oi^ jf^ cfcc., treating a;, y, ibc. as constants^ then any of its invariants will he * In geometry emananta denote the polar cnryes or anrfacesol a point with regarcl to a cmYe or surface. '■ A a covarianl of I as variables. LINE4R TRANSPOEMATIONS. e original quantic when a;, y, &c. are considered We have just seen that x'' -tj + &c. becomes X'' jYr + ^°" when we substitute for x', X^X' + fi,Y' -i- &c., and for a;, X^X + fij Y-i- &c. It ia evidently a matter of indifference whether the substitutions for a/, &c., and for x, &c., are nmultaneous or successive. If then on transforming x', &c. alone, as'' -33 +&c. becomes aX''+&c., then a, &e, will be BDcb fimctionB of x, &c. as when x, y, &c. are transformed will become -ry^ 1 &o. Now an invariant of the given emanant considered as a function of x', y\ £c. only, is by definition such a function of its coefficients as differs only by a power of the modulus from the corresponding function of the transformed coefficients a, 6, &c. But since, as we have seen, a, &c. become ■j=i , &c. when a;, &c. are tranafonued, it follows that the given d'u invariant will be a function of -j-j, , &c., which when a;, &c. are transformed will difi'er only by a power of the modulua from the corresponding function of -j^ , &c. It ia therefore by deGoition a covariant of the quantic. Thus then, for example, since we have proved (Art. 119) that if the binary quantic ax^ + ^bxy + cy' becomes by trans- formation AX* + 2BXY+CY', then 4C-£' = A'(ao-J'); it follows now, by considering the second emanant I x' -r- + 1/ t- 1 « of a quantic of any degree, that " dX' ■ dT' KdXdYj (d^ d^ _ / d'u \*\ \da!''df {dxdi/)y a theorem of which other demonstrations will be given. 127. In general, if we take the second emanant of a qaantlc 1 any number of variables, and form its discriminant, tbia will CONTRA VARIANTS, 117 be a covariant which Is called the Sesstan of the quantic. It was noticed (Art. 118) that the discriminant of every quadratic function may be written as a determinant. Thus then if, as we have done elsewhere, we use the suffixes 1, 2, &c. to de- note differentiation with respect to x^ y^ &c., so that, for example, u^^ shall denote -^ , then the quadratic emanant is u^^af* + 2u^^'j/ + &c., and its discriminant, which is the Hessiani is the determinant &c. &c. ^^ ^^ &c. &c. ^U) **«) ^Vi'i **«!) ^M» ^«8> "ai) ^88) ^ni 128. We have seen (Art. 123) that the determinant of a system of linear equations is an invariant of the system. If then, given a system u, v^ Wy &c of as many functions as variables, we take the first emanants x\ 4 y'w, + z^u^ + &c., &c., their determinant u^, u^j ti,, &c. ^iJ v«) V.) &c. ^11 «^«) ^8» &c. &c. is a covariant of the system. This is the determinant already called the Jacobian (Art. 88). The Hessian is the Jacobian of the system of differentials of a single quantic u^, u,, u,, &c. 129. Contravariants, When a set of variables a;, y, &c. are linearly transformed, it constantly happens that other variables connected with them are also linearly transformed, but by a substitution different from that which is applied to Xj y, &c. If the equations connecting Xy y, z with the new variables be written as before then variables f , 17, (f are said to be transformed by the inverse substitution, if the new variables, expressed in terms of the 118 LINEAR TRANSFORMATIONS. old, are where if in the first substitution the coefficients are the con- stituents of the determinant [\fi^v^ read horizontally, in the second they are the same constituents read vertically; and where if in the first substitution the old variables are expressed in terms of the new, in the second the new are expressed in terms of the old. Stated thus, it is evident that the relation between the two substitutions is reciprocal. Solving for f , ffj ^ in terms of E, H, Z, we get (Art. 29) ac=ah+^sH+-n;z, where X,, J/j, &c are the minors obtained by striking out from the matrix of the determinant (\/Li,v,) (the modulus of transfor- mation) the line and column containing X„ /a^, &c. Sets of variables x^y^ z] f , 17, (f, supposed to be transformed according to the different rules here explained, are said to be contragredient to each other. In what follows, variables sup- posed to be contragredient to a?, y, z are denoted by Greek letters, the letters a, /3, 7 being usually employed in subsequent lessons. We proceed to explain two of the most important cases in which the ipverse substitution is employed. 130. When a function of x^ y^ Zj &c., is transformed by linear substitutions to a function of X, F, Z, &c., then the differential coefficients, with respect to the new variables, are linear functions of those with respect to the old, but are ex- pressed in terms of them by the inverse substitution. We have d ^ d dx d dy d dz ^ dX "" dx dX dy dX dz dX But from the expressions for a;, y, &c. in terms of X, 7) &c.| we have dx ^ d'y _ dz^ ^ dX"^'' dX"^^' dX"^'' Hencethen ^=x,^ + X.| + X,|-h&c. INYEBSE TKANSFOBMATIOMS. 119 Similarly ^=. ;». ^ + ^, ^ + ;», | + &c., &c. Thus then, according to the definition given in the last article, the operating symbols 3~ j j^ ) t" j &c« *^ contragredient to 0?, y^ z^ &c., that is to say, iivhen the latter are linearly trans- formed, the former wiU be linearly transformed also, but according to the different rule explained in the last article. If, as before, u,, u,, &c. denote the differential coefficients of u, and £/j, U^j &c. those of the transformed function Z7, we have just proved that D; = \u^ + \u^ + X3M3, U^ = /A^Wj + /i^u^ + /ligW,, &c. Consequently, If t£„ u^j u^ all vanish, Z7„ Z^, U^ must all vanish likewise. Now we know that u^j u,, u^ all vanish together only when the discriminant of the system vanishes ; if then the dis- criminant of the original system vanishes, we see now that the discriminant of the transformed system must vanish likewise, and therefore that the latter contains the former as a factor, as has been already stated (Art 120). 131. In plain geometry, if a;, ^, z be the trilinear coordinates of any point, and a;f + ^17 + ^(T^ be the equation of any line, f , 17, (f may be called the tangential coordinates of that line (see Conicsy Art. 70). Now, if the equation be transformed to any new system of axes by the substitution a;s\X+&c., the new equation of the line becomes f(\x+/.,r+v,z)+i;(\x+/..r+v,z)+?(\.z+/..r+v3Z)=o, so that if the new equation of the right line be written EZ + H r+ ZZ= 0, we have B = \f + X,i7 + X,C, H = /A,f + /A,i; + A*8Si Z^^f + M + ^.f- In other words, when the coordinates of a point are transformed hy a linear substitution^ the tangential coordinates of a line are transformed hy the inverse substitution; that is, they are con- tragredient to the coordinates of the point. In like manner, in the geometry of three dimensions, the tangential coordi- nates of any plane are contragredient to the coordinates of any 120 LtNEAB TRANSFORMATIONS. point. When we tranaform lo new axes, all coordinatea xj/zw, a'j/Vw', &c. expressing diiFerent points, are cogredient : tbat ia to say, all must be transformed by tlie same substitution x = \X+&c., x' = \X' + &c., &c. But the tangential coordi- nates of every plane will be transformed by the inverse substitution, as we have just explained. Similarly the ray coordinates of different lines for the same system of reference are cogredient, but the axial coordinates are transformed by the inverse substitution, tbat is, are contragredient to the former. See Surfaces, Art. 57e. The principle just stated will be frequently made use of in the form a-f + ^'7 + z?= -^H + yn + ^'Z, where x, y, z being suppoBcd to bo changed by the substitution, x = \X + fi.^T+&e., ^, t}, ^ are supposed to be changed by the inverse substitution H = X,f + X^i; + \^, &c. In other words, in the case supposed, x^ + t/7}-\- efis a function absolutely unaltered by transformation, and analogous statements easily follow in the other cases mentioned. 132. If a function 03;" + &c. becomes by transformation AX' + &c., then any function involving the coefificients and those variables which are supposed to be transformed by the inverse substitutlan is said to be a contravariant if It is such that it differs only by a power of the modulus from the corre- sponding function of the transformed coefficients and variables: that is to say, if 4> {A, B, &c., S, H, &c.) = A'i> {a, 6, &c., f, tj, &o.). Such functions constantly present themselves in geometry. If we have an equation expressing the condition that a line or plane should have to a given curve or surface a relation independent of the axes to which it is referred {as, for ex- ample, the condition that the line or plane should touch the cnrve or surface), then, when we transform to new axes, it ia obviously indifferent whether we transform the given relation by substituting for the old coefficients their values In terms of the new, or whether we derive the condition by the original role from the transformed equation. In this way it is aeea CONCOMITANTS. 121 that the conditions in question are of sacb a kind that ^(a, bj f , &c.) differs only by a factor from if>[A^B^ S, &c.). 133. Besides covariants and contravariants there are also functions involving both sets of variables, which differ only by a power of the modulus from the corresponding trans formed functions :. i.e. such that ^(-4,5,&c,,X,F,&c.,B,H,&c.) = A''0(a,i,&c.,a:,y,&c.,f,i7,&c.). Dr. Sylvester uses the name concomitant as a general word to include all functions whose relations to the quantic are un- altered by linear transformation, and he calls the functions now tinder consideration mixed concomitants. I do not choose to introduce a name on my own responsibility ; otherwise I should be inclined to call them divariants. The simplest ftinction of the kind is ficf + yiy + «f, which we have seen (Art. 181) is trans- formed to a similar function, and is therefore a concomitant of every quantic whatever. 184. K we are given any invariant / of the quantic a^x"" + na^Q^'^y + nh^o^'^z + ^« (n - 1) a,a;""*y' + &c., we can deduce from it a contravarlant by the method used in Art. 123. If a^ -^c &c. becomes by transformation -4qX* + &c., then, since xl^ + &c. becomes XU + &c., it follows that a,aj"+&c. + ifa;f+yi7 + «?f=u4,Z"+&c. + A(ZH + YH+-ZZ)\ Now an invariant of the original quantic fulfils the condition * (A) ^1) ^i) &cO = ^"^ («o» «i) \i &c.). Forming then the same invariant of the new quantic, it will be seen that ^ {A^ + AE*, ^, + AS^-'H, &c.) = A'0 (a„ + if, a, + W^\, &c.)- Since k is arbitrary we may equate the coefficients of like powers of h on both sides of this equation. But, by Taylor's theorem, these coefficients are all of the form B 122 LINEAR TRANSFORMATIONS. We have proved then that they differ only by a power of the modulus from the corresponding function of the transformed equation. They are, therefore, contravariants^ since it is assumed all along that f , 17, f ^^e to be transformed by the inverse sub- stitution. Dr. Sylvester has called contravariants formed by this rule, first, second, &c. evectants of the given invariant. Thus f ** -^— + P^'^f) -J— + &c. is the first evectant It is to be ob- served that in the original quantic the coefficients are supposed to be written wiih^ and in the evectant without^ binomial coeffi- cients. Comparing this article with Art. 123 we see that the function f*-^ — h &c. may be considered either as a contravariant of the single given quantic, or as an invariant of the system obtained by combining with the given quantic the linear func- tion x^ + yq + «?. The theory of contravariants, therefore, may be included under that of invariants. If we perform the operation f * ;7- + &c. upon any covariant we obtain a mixed concomitant, for it is proved in the same way that the result, which will evidently be a function involving variables of both kinds, will be transformed into a function of similar form. Ex. 1. Welmowthatoc— i^isanmvariantof oa;*+2^a?y+cy'j hencecp— 2dfil+ai|» is a contravariant of the same system. Ex. 2. Similarly, abc + 2fgh — af^ — hg'* — cA*, being the discriminant, and there- fore an invariant of ax^ 4- hy^ + cs* + 2/y« + 'igsac + 2^y, is a contravariant of l^e same quantic. Geometrically, as is well known, the function equated to zero expresses the tangential equation of the conic represented by the given quantic. Ex. 3. Given a system of two ternary quadrics aa:» + Ac, a'a* + Ac, then sinoe a* {be —f^) + <&c. is an invariant of the system (Art. 123), we find on operating with P ~ + Ac, that {he' + Vo ^ %ff') ^ + {ca* + c'a - Igg*) n« + {ab' + a'i - 2AA') ^ + 2{gh' + g'h-af'-ay)vt + 2{hf'-\-h'/-bg'--b^g)l^ + 2(fg'+jrg-ch'-e'h)^ is a contravariant of the system. We might have equally found this contravariant by operating with a' -^ + Ac. on the contravariant of the last example. G^metrically, the function equated to zero expresses the condition that a line should be cut har- monically by two conies. EVECTANTS. 123 135. When the discriminant of a qnantic vanishes, it has a set of singular roots x'y'z' [geometrically the coordinates of the double point on the curve or surface represented by the quantic] ; and in this case the first evectant will be the perfect «** power of (a;'f + jy'i; + «'f ). Since we have seen that this evectant is a function unaltered by transformation, it is sufficient to see what it becomes in any particular case. Now if the discriminant vanishes, the quantic can be so transformed that the new coefficients of a;**, aj""*y, q^"^z shall vanish ; that is to say, so that the singular root shall be y = 0, 2; = [geome trically, so that the point yz shall be the double point]. But it was proved (Art. 117) that the form of the discriminant is Evidently then, not only will this vanish when a^, o^, h^ vanisb| but also its differentials with respect to every coefficient except a^ will vanish. This evectant then reduces itself to JT J- multiplied by the perfect n** power f*, which is what (aj'f +y'i7 + «'f )* becomes when j/ and «' = 0, and aj' = 1. Thus then, if the discriminant of a ternary quadric vanish, the quadric represents two lines : the contravariant becomes a perfect square; and if we Identify it with (aj'f 4y'i7+«'(r)', we get aj'yV the coordinates of the intersection of the pair of lines. If a quantic have two sets of singular roots, all the first differentials of the discriminant vanish, and its second evectant becomes the perfect n** power of where a?y«', oi^y^z" are the two sets of singular roots; and 80 on. { 124 ) LESSON XIII. FORMATION OF INVARIANTS AND COVARIANTS. 1S6. HAVlNa now shewn what is meant by invariants, &c., we go on to explain the methods bj which such functions can be formed. Three of these methods will be explained in this Lesson, and a fourth in the next Lesson. Syrnmetrio functicna. The following method is only appli- cable to binary quantics. Any symmetric function of the differences of the roots is an invariant^ provided that each root enters into the expression the same number of times.* It is evident that an invariant must be a function of the differences of the roots, since it is to be unaltered when for x we substitute as + X. Now the most general linear transformation is evidently eqni- valent to an alteration of each root a into r-? > • By this change the difference between any two roots a — /S becomes A/^ "" /x^N/\o ■ — \\ • lo ^^^^^ then, that any function of the differences may, when transformed, differ only by a factor from its former value, it is necessary that the denominator should be the same for every term ; and therefore the function must be a product of differences, in which each root occurs the same number of times. Thus for a biquadratic, S (a — /S)* (7 — S)* is an invariant, because, when we transform, all the terms of which the sum is made up have the same denominator. But * If in the equation the highest power of a; is written with a coe£&cient o^ we have to divide b^ that coe^cient in order to obtain the e^ipression for the snin, Ac of the roots ; and all symmetric functions of the roots are fractions containing poweis of Oo iu the denominator. When we say that a symmetric function of the roots is an invariant, we .understand that it has been made integral by multiplying it by such a power of a, as will clear it of fractions ; or, what comes to the same thing) if we form the symmetric function on the supposition that the coe^cient of x" is 1, that we make it homogeneous by multiplying each term by whatever power of a« may be necessary. FROM THE ROOTS OP BINARY QU ANTICS. 125 S{a — /9)' is not an invariant, the denominator for the term (a - /3)" being {\'a + /)* (^'i^ + /)") a°d for the term (7 - Sy being (V7 + /)»(VS + /)*. 137. Or perhaps the same thing may be more simply stated by writing the equation in the homogeneous form. We saw (Art. 120) that if we change x into \x + fiy^ y into Va? + fi!y^ the quantity x^y^ — xjf^ becomes (X^'- V/a) (o?^, — a^^yj, and consequently any function of the determinants x^y^ - x^y^ &c. is an invariant. Now (Art. 61) any function of the roots expressed in the ordinary way is changed to the homogeneous form by writing for a, /9, &c. — , — , &c., and then multiplying by such a power of the product of all the y's as will clear it of fractions. If any function of the differences in which all the roots do not equally occur be treated in this way, powers of the ^'s will remain after multiplication, and the function will not be an invariant. Thus, for a biquadratic, 2(a — /3]' be- comes 2yg'y/(a?,y, — a?j,y,)*; but the function S (a — ^"(7 — 8)", in which all the roots occur, becomes 2 {x^^ — ^ayj* (^s^* "" ^JH^% and this being a function of the determinants only, 18 an invaridnt. It is proved in like manner, that any symmetric function formed of differences of roots and differences between x and one or more of the roots is a covariant, provided that each root enters the same number of times into the expression. Thus for a cubic S (a — ^)* [x — 7)* is a covariant. 138. We can, by the method just explained, form invariants or covariants which shall vanish on the hypothesis of any system of equalities between the roots. Thus, let it be required to form an invariant which shall vanish when any three roots are all equal, it is evident that every term must contain some one of the three differences a — )8, /9 — 7, 7 — a; and in like manner for every other set of three that can be formed out of the roots. Thus, in a biquadratic, there are four sets of three roots: the difference a - ^ belongs to two of these sets, and 7 — S to the other two ; therefore 2 (a - fif (7 - 8j** is an invariant which * Z (a — j3) (y — ^) would vanish identically. 126 FORMATION OF INVARIANTS AND COVARIANTS. "Will vanish if any set of three roots are all equal. In like manner, for a quintic there are ten sets of three : a — /3 belongs to three sets, 7 — S to three other sets ; the remaining sets are 078, aSs, ^rytj /88s, two of which contain 7 — e and the other two S - e. The function then 2 (a - /8)* (7 - S)' (S - e)' (7 - e)" is an invariant which will vanish if any set of three roots are all equal. This invariant (Arts. 57, 58) is of the fourth order and its weight is 10. So, again, if we wish to form a covariant of a biquadratic which shall vanish when two distinct pairs of roots are equal, the expression must contain a difference from each of the pairs a-/9, 7 — S; a — 7, ^^ S] a-S, ^ — 7. Such an expression would be s(a-/3r(/3-7r(7-«r(«^-8r, or 2(a-/3)(a-7)(a-S)(a:-^)»(aj-7)»(x-8)", which are covariants of the fourth and sixth degrees respectively in the variables ; and of the fourth and third in the coefficients, and every term of each vanishes when two distinct pairs of roots are equal. 139. Mutual differentiation of covariants and contravartants. When we say that <f> (a, J, f , 17, &c.) Is a contravariant, f , 17, &c. may be any quantities which are supposed to be transformed by the reciprocal substitution. Now we have shewn (Art. 130) that the differential symbols -j~ ^ -j- 1 &o. are so transformed. We may, therefore. In any contravariant substitute these differ- ential symbols for |, 17, &c., and we shall obtain an operating symbol unaltered by transformation, and which, therefore, if applied either to the quantic itself or to any of Its covariants, will give a covariant If any of the variables remain after differ- entiation ; and if not, an Invariant. Similarly, If applied to a mixed concomitant. It will give either a contravariant or a new mixed concomitant, according as the variables are or are not removed by differentiation. Or, again, in any contravariant in- stead of obtaining an operating symbol by substituting for f, 17, &c., ;7- ) 7- J &c., we may substitute -7— , -^, &c, where IN ANY NUMBER OP VARIABLES. 127 U Is either the quantic Itself or any of Its covariants, and so obtain a new covarlant. The relation between the sets of vari- ables a?, y, «, &c., f , 17, f, &c. being reciprocal, we may, in like ,, • r> Q d d d ff manner, substitute m any covariant,for a?,y,;2J,&c., -rz^^ ^ ~ru>j &c., when we get an operative symbol which when applied to any contravariant will give either a new contravariant or an In- vainant. Thus then. If we are given any covarlant and contravariant, by substituting in one of them differential symbols and operating on the other, we obtain a new contravariant or covarlant ; which again may be combined with one of the two given at first, so as to generate another ; and so on. 140. In the case of a binary quantic, this method may be stated more simply. The formulae for direct transformation being those for the reciprocal transformation are (Art. 129) whence Af = fi^S — X^H, A17 = - /i,B -f XjH, which may be written Ai7 = \H + /*,(-E); A(-?)=\H + ;*J-H). Thus we see that, with the exception of the constant factor A, f) and — ^ are transformed by exactly the same rules as X and y; and It may be said that y and — x are contragredlent to X and y. Thus then, in binary quantics, covariants and coutravariants are not essentially distinct, and we have only In any covarlant to write rj and ~ f for a? and y, when we have a contravariant, or vice versd. In fact, suppose that by trans- formation any homogeneous function whatever (a?, y) becomes 4>(X, y), the formulae just given shew that (f>{rjj — f) will become A~^^ (H, — E), where p is the degree of the function in X and y. If then 0(a;, y) is a covariant, that is to say, a function which becomes by transformation one differing only by a power of A from a function of like form in X and Y", evidently <j> (17, — f ) will by transformation become one differing only by 128 POBMATION OF INVARIANTS AND COVARIANTS. * a power of A from one of like form in H and H ; that Is to say, it will be a contravariant. For example, the contrayariant, noticed (Art. 134, Ex. 1), c|' - 2 Jfi; + aiy*, by the substitution just mentioned, becomes the original quantic. Instead then of saying that the differential symbols are contragredient to x and y, we may say that they are cogredient to r/ and — x ; and if either in the quantic itself or any of its covariants we write -r- j — jr- for a? and y, we get a differential symbol which may be used to generate new covariants in the manner explained in the last article. Or we may substitute -r , — -- for 0? and y, and so eet a new covariant. The dy^ dx ^' ^ following examples will sufficiently illustrate this method : Ex. 1. To find an inyariant of a qnadratic, or of a systein of two qnadttiticfl. Suppose that by transformation oo* + 2bxy + cy* becomes AX^ + 2BXr+ CF*, them d d since we hare seen that A j- , — A ^ aire transformed by the same rules as k and y, it follows that the operative symbol If then we operate on the given qnadratic itself, we get 4A2 {ac - 62) = 4 ^AC- JB«), which shows that ae — 6^ is an invariant ; or if we operate on a*ai^ + Wxy + ffy^ and the transformed function, we get 2 A2 {cu^ + ca' - 2^0 = 2 {AC + CA* - ^BBTj, which shews that ac* + ca* — 266' is an invariant. We might also infisr that a {bx + eyf - 26 (6a? + ey) {ax + 6y) + c (aa? + by^ is a covaiiant ; but this is only the quantic itself multiplied by a<; — h\ Ex. 2. Every binary quantic of even degree has an invariant of the second order in the ooefELcients. We have only to substitute, as just explained, ^ , — ^ for cb and jf, and operate on the quantic itself. Thus for the quartic (o, 6, c, rf, e^f a?, y)* we find that ae — 46<i + 3c2 is an invariant; or for the general quantic {a^ ai...a«.j, Ot^x, y)*, we find that a^On — na^On-i + in (» — 1) Oj^^ — Ac. is an invariant ; wheie the coeffi- cients are those of the binomial, but the middle term is divided by two. If we apply this method to a quantic of odd degree ; as, for example, if we operate on the cubic ax^ ■¥ ^hxh, + ^cxy^ + dy^, with rf_-3c^^ + 36 ^^- a^, it will be found that the result vanishes identically. We thus find, however, that a i^tem of two cubics has the invariant {acP — a'd) — 3 {be' — b'c). Or, in general, that a system of two quantics of odd degree, a^ + &c , 60a:* + Ac, has the invariant (aobn - Onbo) - n (ai6„-i - an-M + hn{n- 1) (0,6,-2 - a»-M &c., which vanishes when the two quantics are identical. BY MUTUAL DIFFERENTIATION. 129 141. When, by the method just explained, we have found an invariant of a quantic of any degree, we have immediately, by the method of Art. 126, a covariant of any quantic of higher degree. Thus, knowing that ac - b* is an invariant of a quad- ratic, by forming that invariant of the quadratic emanant of , , d^u d*u / d*u \* . any quantic, we learn that -r-y -^ - f , , I is a covanant of any quadtic above the second degree. In like manner, from the invariant of a quartic oc- 4Jc?+ Sc*, i^e infer that for every quantic above the foiirth degree d^u d*u , d^u d^u ^ ( d u -4.r-r7--r-n+3 / d'u y dx* ^ dx^dy dxdy* \da?dy^ is a covariant, &c. In this way we see that a qdaiitic in general has a series of covariants, of the second Order in the coefficients, and of the orders 2(n — 2), 2(n — 4), 2(n — 6), &c. ill the variable]^. Those covariants may be combinod with the original quantic and with each others so as to lead to new co^ Variants or invariants; Ex.. 1. A qnartdc has an inyariant of the third order in the' coef^enta. We knoW that its Hessian (oa? + V»xy + <>rt (c»» + 2<fey + <^ - (fee* + 2exy + dy'*f, or (oc - *«)a:*+ 2 (a<l - ftej) aj»y + (fl* + 2W - 8c«) a^y + 2 (ft6 - c<i) ajy* + (ce - d*) y«, is a oovadaat. Operate on this with {a, b, e, d, ej^r- , — 7 ) , and we get setenly- two times ack + 2&<i - <wi« - 66« - c», #hich is tUffefore an imrariant. Ex. 2. Eveiy quantic of odd degree has an inyariant of the fourth order in the ^beffldents. The qnantk has a qaadaiiib ooVaiiabt ^|^ TXT ~ ^^* ^^ ^^ second order in the coefficients ; atid the diserimifiant of this cfnadtatic wiU be an invariant of the original qnantic (Art. 124), and will be of the fotirth order in its coeffidefits. In fact, It is prored in this v^&y that every qnantic has an invariant of the fourth order; for if we take any of the dorftriants of this article, which are all of even degiee, its invariiant of the second order will be of the fourth order in the coefficients of the original qoatftic. But wheri the quantic is of even degiee, it may happen that the invariant so found is only the square of its invariant of the second order. Ei. 8. To form the invariant of the fourth order for a cubic. Its Hessian is (oa? + iy) (ex + dy) - (6a; 4 cy)* j or {ac --b^ 3c^ + {ad- be) a:y + (W - <?«) p\ a 130 FORMATION OP INVARIANTS AND COVARIANTS, Hence {ad - bef -4{ac-b^ {bd - c^ is an invariant of the cubic. In fact, it is its discriminant ««d« - Qabcd + 4ac» + Ui^d - W^c\ 142. From any invariant of a binary quantic we can gene- rate a covariant. For from it we can form (Art. 134) the evectant contravariant |*^— + &c. ; and then in this substi- tuting y, — a? for f and 17, we have a covariant. For example, from the discriminant of a cubic which has been just written we form the evectant f ' (ad* - 36crf + 2c») + 3f 17 (- rtcti + 25^- Jc») 4 3fi7'(-aJrf+2ac'-J'c) + i7*(aV-3aSc + 2J'), whence we infer that the cubic has the cubic covariant (aV-3a Jc+2 J', ahd- 2ac"+ J*c, -acrf+2iV- hc\ Sbcd- ad'-2c'3[aj,y)'. 143. The differential equation, — We saw (Art. 62) that in- variants satisfy certain partial differential equations, and these furnish a third method of forming these functions based on the following principle. If n be the order of a binary quantic^ the order in the coefficients of any of its invariants^ then the weight (see Art. 56) of every term in the invariant is constant and = i^O. For if we alter x into \x^ leaving y unchanged, since this is a linear transformation, the invariant must, by definition, remain unaltered, except that it may be multiplied by a power of X, which is in this case the modulus of transformation. It is proved then, precisely as in Art. 57, that the weighty or sum of the suflixes, in every term is constant. Again, the invariant must remain unaltered, If we change X into y, and y into x^ a linear transformation, the modulus of which is - 1. The effect of this substitution is the same as if for each coefficient a^ we substitute a . Hence the sum of a number of suffixes a + )8 + 7 + &c. = (n — a) + (n — /8) + (n — 7) + &c., whence 2(a + /8 + 7 + &c.) = w^. q.e.d. Cor. n and 6 cannot both be odd, since their product is an even number ; or, a binary quantic of odd degree cannot have an invariant of odd order. BY THE PARTIAL DIFFERENTIAL EQUATION. 131 144. The principle just established enables us to write down immediately the literal part of any invariant whose order is given. For the order being given, the weight is given also. Thus, if it were required to form for a quartic an invariant of the third order in the coefficients, the weight must be 6, aqd the terms of the invariant must be where the coefficients -4, 5, &c. remain to be determined. The reader will observe that there are as many terms in this in*- variant as the ways in which the number 6 can be expressed as the sum of three numbers from to 4 inclusive ; and gene- rally that there may be as many, terms in any invariant as the ways in which its weight \n6 can be expressed as th^ sum of 6 numbers from to w inclusive. We determine the coefficients from the consideration that since an invariant is to be unaltered by the substitution either of aj + \ for ar, or of y + \ for y, evidently, as in Art. 62, every invariant must satisfy the two differential equations dl dl dl ^ dl , ^ dl o it being supposed that the original equation has been written with binomial coefficients. In practice only one of these equa* tions need be used ; for the second is derived from the first by changing each coefficient a^ into fl^_„. It is sufficient then to use one of the equations, provided we take care that the func- tion we form is symmetrical with regard to x and y\ that is to say, does not change (or at most changes sign)* when we change a^ into a^.^. "~^And this condition will always be fulfilled if we take care that the weight of the invariant is that which has been just assigned. Thus then, in the example chosen for an illustration, if we operate on Aa^a^a^ + &c. with ♦ When we change x into y and y into x, this is a transfonnation whose modulus is 0, 1 1,0 or — 1. Any invariant, therefore, which when transformed becomes multiplied by an odd power of the modulus of transformation will change sign wheii we interchange x and y. Such invariants are called skew invariants. 132 POKMATJON OF INVABIANTS AND CuVARIANTS. «o^ + &c., we get {2B + 2 A ) a^a^a^ + (D + 6 (7 + 4^) a/i^a^ + (22) + 45) a,a,a, + (6£+ 3i)) a,a,«, = 0, wbence If we take A = ly the other coefficieDts are found to be -B= - 1, i? = 2, C= - 1, ^= - 1, and the invariant is 145. In seeking to determine an invariant of giv^p o^^d^r ]>j the method jnst explained, we have a certain number of unknown coefficients A^ B^ (7, &c. to determine, {^nd we dp so by the help of a certain number of conditions formed by means of the differential equation. Now, evidently, if the number of ^hese conditions were greater than the number of unknown ^e^cients, the formation of the invariant would In general be Impossible^ If they were equal we could form one Inyarlant;. If the number of conditions were less, we could form more than one Invariant of the given order. We have just seen that the number of terms In the Invariant, which Is one more than the number of unknown coefficients. Is equal to the number of ways In which its weight \n6 can be written, as the sum of 6 numbers, none being greater than n. But the effect of ..... the operation a^-^ — V &c. Is evidently to diminish the weight by one, the number of conditions to be fulfilled Is, therefore, equal to the number of ways in which \n6 — I can be expressed as the sum of 6 numbers, none exceeding n. Thus, In the example of Art. 144, the number of conditions used to deter^ mine A^ Bj &c. was equal to the number of ways In which 5 can be expressed as the sum of three numbers from to 4 Inclusive. To find then generally whether an invariant of a binary quantic of the order can be formed, and whether there can be more than one, we must compare the nuniber of ways in which the numbers ^nd^ ^nd- 1 can be expressed as the sum of 6 numbers from to w inclusive.* * It was in this way Prof. Cayley first attempted to investigate the nnmber of invariants and covariants of a binary quantic. BY THE PARTIAL DIFFERENTIAL EQUATION. 133 146. Similar reasoning applies to covariants. A covariant, like the original quantic, must remain unaltered, when we change x into pXj and at the same time every coefficient a^ into p^a^. If then the coefficient of any power of Xj xf*" in the covariant be a*J^Cy, &c., it is obvious, as before, that /A + a + )8 + &c. must be constant for every term ; and we may call this number the weight of the covariant. Again, in order that the covariant may not change when wc alter x into y and y into x, we must have /* + a + )8 + 7 + &c. =:(^-/A) + (n-a)-f(n-)8) + &c., where p is the degree of the covariant in x and y ; whence if be the order of the coyariant in the coefficients, we have immediately its weight =i {nO^-p). Thus if it were required to form a qpadratic covariant to a cubic, of the secpnd order in the coefficients, w = 3, ^=^ = 2, and the weight is 4. We have then for the terms multiplying a?*, a + i8 = 2, and these terms must be aji^ and a^^. In like manner the terms mulr tiplying icy must be ajx^^ a^a^y and those multiplying y* must be ajflj, afi^. In this manner we can determine the literal part of a covariant of any order. The coefficients are determined as follows : 147. From the definition of a covariant it follows that we must get the same result whether in it we change x into x H- Xy, or whether we make the same change in the original quantic and then form the covariant. But this change in the original quantic is equivalent (Art. 62) to changing a, into a, + a^X, a^ into a, + 2a^X + a^X", &c. Hence, in the covariant also the change of a; to a; + Xy mufit be equiyalent to changing a^ into a^ + a^X, &c. Let the covariant then be A^af -{-pA^any + ^p (i> - 1) A^a^'*^ + &c. Let us express that these two alterations are equivalent, and let us confine our attention to the terms multiplying X. Thei^ if, as in Art. 64, we use as an abbreviation to denote the operation a^ -^ + 2aj -^ — I- &c, the symbol -rp, we get 134 FORMATION OP INVARIANTS AND COVARIANTS. In like manner, writing -j- for wa, 7— + (n — 1) a^ -^ — I- &c., we have • * dA Thus we see that when A^ is determined so as to satisfy -^ = ; in other words, when A^ is a function of the differences of the roots of the quantic (Art. 58), all the other terms of the covariant are known. The covariant is in fact It will be observed that the weight of the covariant being ^(wff + p) the weight of the term A^ is i(w^-p), since the weight of A^ together with p makes up the weight of the covariant. This term A^^ whence all the other terms are de- rived, was named by Prof. M. Roberts the source of the covariant. He observed also that the source of the product of two cova- riants is the product of their sources. For if we multiply the covariant last written by we get, as may be easily seen, Hence, if we know any relation connecting any functions of the differences A^^ B^^ C^^ &c., the same relation will connect the covariants derived from these functions. Ex. 1. To find the quadratic covariant of a cubic. We have seen (Art. 146) that A^ is of the form o^q + Baya^. Operate on this with a^ ^ — V 2ai -z — , and we get (2 + 2B) affli = 0, whence 5 = — 1 and A^ = a^^ — a^ai. Operate then with Soj -z — h 2^2 ;j — *" *^3 T" » ^^^ ^® ha.YQ 2 Ay = ajO^ - a^i. Operate with the same ddn Udy CLCL^ on Ai, and we have A^ = a^a^ — agaj- The covariant, therefore, is {a^Q - ttiQi) x^ + {a^ai - a^i) xy + [a^a^ - a^a^ y\ LAW OP COEFFICIENTS OF BINARY COVARIANTS. 135 Ex. 2. To find a cubic covariant of a cubic of the third order in the coefficients. Here n = 3, 6 = 3, ^ (nd +p) =z6. The sum then of the suffixes of the coefficient of a^ wiU be 3 ; and this coefficient must be of the form Aa^^fl^ + Ba^ia^ + Ca^a^a^, d d d Operate with a^ ^ — h la^ j- + 303 -5— , and we get 1 ^^^3 • (3^ + B) a^^a^ + (25 + 3C) a^a^a, 0) whence if we take j1 = 1, we hare B = — 8, C=2jOt Aq = a^a^^ — Sa^iO^ + 2a|aia,. J J J Operate on this three times successively with 3«i -r- + 203 -^ — h a^ -j— , and we have the remaining coefficients, and the covariant is (see Art. 142) (os^oflfo — ^o.fixO'^ + 2aiaiOi) a* + 3 {a^a^a^ — 2a^a^^ + Os^iAi) aj'y + 3 {2a^iai — OjOgai — a^ot^) xy^ + (SosOsAi — 20)0^ -^ ^^t^) 9** 148. We have seen that a quantic has as many covariaDts of the degree p in the variables and of the order in the coefficients as functions A^, whose weight is ^ [nO — ^) can be found to satisfy the equation -^ = 0. And, as in Art. 145, we see that this number is equal to the difference of the ways in which the numbers ^ [nd-^p) and ^ [nO ^p) — 1 can be expressed as the sum of numbers from to n inclusive. It may be re- marked that p cannot be odd unless both n and are odd. Hence only quantics of odd degree can have covariants of odd order in the coefficients, and these must also be of odd degree in the variables. 149. The results arrived at (Art. 147) may be stated a little differently. The operation y -r- performed on any quantic is equivalent to a certain operation performed by differentia- ting with respect to the coefficients. Thus, for the quantic (or^, a„ Og...3[a;, y)", we get the same result whether we operate on it with y -j- or with a^-j — '" ^^1 ^7 — *" ^^' ^^^^ latter opera- tion then may be written ^ ;t- ; and the property already proved for a covariant may be written that we have for it y -j y ~ = 0. In other words, that we get the same 136 FOKMATION OP INVARIANTS AND COVARIANTS. A result whether we operate on the coyarlant with y-7- or with d d €^ -J — h 2a^ -= — h &c. In his Memoirs on Quantics, Prof. Cayley has started with this propiBrty as his definition of a covariant ; a definitioh which includes invariants also, since for them we have y — = 0, and therefore also y ;t- =0. 150. It call be proVed, in like maimer, that covariants of qtiantics in any number of variables satisfy differential equations which may be written y ^ = [y|^] , ^"^ = [^^] , &c- Thus, for the quantic (a, J, c, j^ g, AJoj, y, i)", we have d d d ^T d d d ^. d ^ d and everj isotariant mnst satisfy these two equations; While dvery invariant most satisfy the two equations dl dl „.di „ dl .dl ^ dl . «^+^^ + '*^ = ^' «^ + *^ + '^^=^5 as may easily be proved from the consideration that the iiitaiiant remains unaltered if T^e substitute for x^x^-'Ky ot x^- (is. t 137 ) LESSON XIV. SYMBOLICAL BBPEBSENTATION OP CONOOMITANTa 151. It remains to explain a fourth method of finding invariants and covariants, given by I^rof. Cayley in 1846 {Cambridge and Dublin Mathematical Journal^ vol. I. p. 104, and Grellej vol. xxx.) ; which not only enables us to arrive at such functions, but also affords the basis of a regular calculus by means of which they may be compared and identified. Let a?j, y^ ; a?,, y, be any two cogredient sets of variables ; then, if we write briefly for _, _, ^; f^, ,y^, f^, &c., it has been proved (Arts. 130, 120, 139) that f„ i;^, f^, tj^ are transformed by the reciprocal substitution; that f,i7j — fgiJi is an invariant symbol of operation ; and that if we operate with any power of this symbol on any function of a;,, y^, a?,, y„ we obtain a covariant of that function. We shall use for {^17, - ^^rf^ the abbreviation 12. Suppose now that we are given any two binary quantics Uj F, we can at once form covariants of this system of two quantics. For we have only to write the variables in U with the suffix (1), those in V with the suffix (2), and then operate on the product UV with any power of the symbol 12 ; the result must be an invariant or covariant. Thus if we operate . « . 1 —— •■•IT 1 • du dr dU dr « • 1 simply with 12 we obtain the Jacobian -7- -t -j— 3—, which *^ ax ay ay ax we saw (Art. 128) was a covariant of the system of quantics. Again, let £/-= ax^ + 2hx, y, + cy,' ; F= a'x^ + ib'x^y^ + c'y,*, then if we operate on TJV with 12*, which, written at full length, is the result is ac'-i^ca' — 2jy, which is thus proved to be an Invariant of this system of quantics. In general, it is obvious T '{ 138 SYMBOLICAL REPRESENTATION OP CONCOMITANTS that the diflferentlals marked with the suffix (1) only apply to U^ and those with the suffix (2) only to F; and it is u^ecessary to retain the suffixes after differentiation ;^ so that 12^ applied to two quantics of any degree gives the covariant dx* rfy* dy* dx^ dxdy dxdy* Similarly the symbol 12' applied to two cubics gives the invariant or to any two quantics gives the covariant d^Ud^V , d^U d'V , d'U d'V d'Ud^V dx* dy* dx*dy dxdy* dxdjf dx?dy dy^ da? ' and so in like manner for the other powers of 12. 152. We can by this method obtain also Invariants or co- variants of a single function U. It is, in fact, only necessary to suppose in the last Article the quantics 27 and V to be identical. Thus, for instance, in the example of the two quadratics given in the last Article, if we make a^a\ J = i', c = c', the invariant 12* becomes 2 [ac — V\ And, in like manner, the expression there given for the covariant 12* of a system ?7, F, by making 17= F, gives the covariant of a single quantic €b? dy^ \dxdy) * In general, whenever we want by this method to form the covariants of a single function, we resort to this process : — We first form a covariant of a system of distinct quantics, and then suppose the quantics to be made identical. And in what follows, when we use such symbols as 12" &c. without adding any subject of operation, we mean to express derivatives of a * If TF be any function containing x^, yi ; a;2) fft i '^'^ S^ the same result whether we linearly transform these variables, and afterwards omit all the suffixes in the transformed equation ; or whether we omit the suffixes first, and afterwards transform X and y. This results immediately from the fact that x^ y^; o^, y^; x, y are cogredient. It follows then at once that if PT, written as a function of x^ yt; x^ytt be a covariant of Uj V; that is to say, if the expression of the coefficients of TF in terms of the coefficients of U and V be unaffected by transformation, then W is abo 9 covariant when the suffixes are all omitted. FOR ANY NUMBER OF FUNCTIONS. 139 Single function K We take for the subject operated on the product of two or more quantics t/j, 0^, &c., where the variables a?j, y, ; a* , y^ ; &c. are written in each respectively, instead of X and y; and we suppose that after differentiation all the suffixes are omitted, and that the variableS| if any remain, are all made equal to x and y. 153. From the omission of the suffixes after differentiation! it follows at once that it cannot make any difference what figures had been originally used, and that 12"* and 34"* are expressions for the same thing. In the use of this method we have constantly to employ transformations depending on this obvious principle. Thus, we can show that when n is odd, 12* applied to a single function vanishes identically. For, from what has been said, 12'*s21'*; but ]2 and 21 have opposite signs, as appears immediately on writing at full length the symbol for which 12 is an abbreviation. It follows then that 12"* must vanish when n is odd. Thus, in the expansion of 12^*, given at the end of Art. 151, if we make U= F", it will obviously vanish identically. The series 12^, 12^, 12^, &c. gives the series of invariants and covariants which we have already found (Art. 141). It is easy to see that, when n is even, 12*" applied to (a^, a^, a,...Xa?, y)" gives a^a^ - na^a^^ + Jn (n - 1 ) a,a^, - &c., where the last coefficients must be divided by two, as is evident from the manner of formation. In particular, we thus have the invariants, for the quadratic, (w — V*] for the quartic, ae - 4W + 3c* ; for the sextic, ag — Qbf+ l5oe — lOd* ; and so on. 154. The results of the preceding Articles naturally extend to any number of functions. We may take any number of quantics 27, F, Wj &c., and, writing the variables in the first with the suffix (1), those in the second with the suffbc (2), in the third with the suffix (3), and so on, we may operate on their product with the product of any number of symbols 12% 23^, 31'', 14', &c., where, as before, 23 is an abbreviation for fjiJa^fs^s) ^®* -^ft^r the differentiation we suppress the suffixes, and we thus get a covariant of the given system of 140 STSIBOLICAL REPRESENTATION OP 00NCOMITAKT3 quantics, which will be an inTariant if it happens that no power of X and y appear after dlffereutiation. Any number of the quantica U, V, W, &c., may be identical ; and in the case with which we shall be moat fi-equently concerned, viz., where we wish to form derivativea of a single quantic, the subject operated on is U^U^U, &c,, where (7, and U^ only differ by having the variabiea written with different suffixes. It is evident that in this method the order of the derivative in the coefficients will be always equal to the number of different figures in the symbol for the derivative. For if alt the functiona were distinct, the derivative would evidently contain a coefficient from every one of the quantica ?7, F, JV, &c- ; and it will be BtiU true, when U, V, IV are supposed identical, that the degree in the coefficients ia equal to the number of factors in the product U^n^U^ &c., which we operate on. Thus the derivatives cout- sidered in the last Article being all of the form 12' are of the second order only in the coefficients. Again, if it were required to find the degree of the derivative in X and y. Suppose, in the first place, that the quantica were distinct, U being of the degree n, Voi the degree n', W of the degree n", and so on ; and suppose that in the operating symbol the £gure 1 occurs a times ; 2, ^ times ; and ao on ; then, since U is differentiated a times, F, |S times, Sic, the result is of the degree (n - a) + («' - ^) + (n" - 7) + &c. When the quantics are identical, if there are p factors in the product VJJ^...1J„ which we operate on, the degree of the result in x and y will be ?i^— (a + (3 4 7 + Ac). While again, if there be r factors such as 12 in the operating symbol, it ia obvious that (a + /3 + 7 + &c.) = 2r. It is clear that if we wish to obtain an invariant, we muat have a = ff = y = n. 155< To illustrate the above principles, we make aa ex- amination of all possible invariants of the third order in the coefficients. Since the symbol for these can only contain three figures, its most general form is 12°. 23''. 31'; while, in order that it should yield an invariant, we muat have a + 7 = « + ^ = /3 + 7 = n, whence a = /3 = 7. The general form, then, that we have to INVARIANTS OF THE THIRD ORDER. 141 examine is (12.23.31)*. Again, if a be odd, this derivative vanishes identically; for, asJn^Art. 153, _by_mterchanging the figures 1 and 2, we have (12.23.31)*- (21.13.32)*; but these have opposite signs. It follows, then, that all mvariants of the third order are included in the formula (12.23*31)*, where a is even. Thus, i2".23*.3i* is an invariant^of ^ quartic, since the differentials rise to the fourth degree; 12*.23*.31* is an invariant of an octavic ; 12*.23*.3l* of a quantic of the twelfth degree, and so on ; only quantics whose degree is of the form 4971 having invariants of the third order in the coefficients. _If we wish actually to calculate one of these, suppose 12*.23^.31', write, for brevity, f„ i;,, &c, instead ^^ 3~ i jr"J ^^'i *^^ ^® have actually to multiply out * ^' In the result omit all the suffixes, and replace ^ by -^-4 &c. ; or, when we operate on a quartic, by a^ the coefficient of a?*, &c There are many ways which a little practice suggests for abridging the work of this expansion, but we do not think it worth while to give up the space necessary to explain them; and we merely give the results of the expansion of the three invariants just referred to. 12*.23''.31' yields the invariant of a quartic already obtained (Art. 141, Ex. I, and Art. 144), viz. :^ r2*.23*.31* gives + «6 (3«6^o" ^^5^1 "• 22a^a,+ 240,^3) +a,(24aga,-36a^aj + ISajaja^. And 12*.23*.Fl* gives «w(«6^o-6«6^i+ 15«A- lH«8)+«ii(-K«o+ 30a,a -54a^a,H-30a^o,) + a,^ (ISa^a^ - 54aya, + 24a,a, + ISOa^a, - 135a^aJ + a, (- lOa^a^ + SOa^^ + 150a,a, - 430a,a, + 270a,aJ + a^ (- 13503^, + 270a,a3 + 4:d5ajni^ - ^^Oa^a^ + a, (- 540a,a, + 720a3a3) - 280a3a3a3. 156. Though the above-mentioned is the only type of invariants of the third order, there is an unlimited number of 142 SYMBOLICAL REPRESENTATION OF CONCOMITANTS. eovariaDts, the simplest being 12^*13, which, when expanded, is dj^ dy* dy dx^dy \ dxdy dy dy* dx ) d^U fd^dU^^ d^ dU\d^dnrdU dxdy* \dx* dy dxdy dx) dy* da? dx * When this is applied to a cubic, it gives the evectant obtained already (Art. 142). The general type of invariants of the fourth order in the coefficients is (12.34)* (T3.24)^ (14.23)^ Thus the discriminant of a cubic is expressed in this notation as (12.34)^(13.24); but we cannot afford space to enter into greater details on this subject. 157. The principles just laid down afford an easy proof of a remarkable theorem first demonstrated by M. Hermite, and to which we shall refer as " Hermite's Law of Reciprocity." The number of invariants of the n** order in the coefficients possessed hy a binary quantic of the p^ degree is equal to the number of invariants of the order p in the coefficients possessed by a quantic of the n** degree. We have already proved that if any symbol 12^23^34^ &c. denotes an invariant of the order p of a quantic of the degree n, then the number of different figures 1, 2, 3, &c., 13^, and each figure occurs n times. But we might calculate by the method of Art. 136 an invariant S (a-/3)" (/3-7)^(7-8)'&c., where we replace each symbol 34 by the difference of two roots (7 "" ^)* T^^s latter is an invariant of a quantic of the p^ degree, since there are by hypothesis p roots ; and it is of the order n in the coefficients of the equation (Art. 58). Thus, for example, a quadratic has but the single independent invariant (a — )8)'*, though of course every power of this is also an invariant ; and the general type of such invariants is (a — ^f^. . Hence, only quantics of even degree have invariants of the second order in the coefficients, and the general symbol for such invariants is 12*^. So again, cubics have no invariant except the discriminant (a - fiY {13 - 7)* (7 - a)* and its powers ; and the discriminant is of the fourth order in the coefficients. Hence, only quantics of hermite's law of reciprocity. 143 the degree Am have cubic invariants whose general type is 12**.23*"'.3i'". It will be proved that quartics have two inde- pendent invariants, one of the second and one of the third order, in the coefficients; and, of course, any power of one multiplied by any power of the other is an invariant. HencCi quartics have as many invariants of the p*^ order as the equation 2x + Sy=p admits of integer solutions; this is, therefore, the number of invariants of the fourth order which a quantic of the p*^ degree can possess. 158. Hermite has proved that his theorem applies also to covariants of any given degree in x and y ; that is to say, that an n^ possesses as many such covariants of the jp>*^ order in the coefficients as a p*" has of the n** order in the coefficients. For, consider any symbol, 12^.23'*.34'' &c., where there are p figures, and the figure 1 occurs a times, 2 occurs b times, and so on ; then we have proved that the degree of this covariant in x and y is (n — a) + (n — i) + &c. But we may form the symmetric function S (a - /3)^ {15 - jY (7 - «)' {^ - «r (« - /3)""* &c., which has been proved (Art. 137) to be a covariant of the quantic of the ^** degree, whose roots are a, ^, &c. Every root enters into its expression in the degree w, which is there- fore the order of the covariant in the coefficients, and it obviously contains x and y in the same degree as before, viz, (w — a) •+ (n — J) + &c. Thus, for example, the only covariants which a quadratic has are some power of the quantic itself multiplied by some power of its discriminant, the general type of which is the order of this in the coefficients is 2p + ^, and in x and y is 2q. Hence we infer that every quantic of the degree 2p + q has a covariant of the second order in the coefficients, and of the degree 2q in x and y^ the general symbol for such covariants being 12^*. When g = 1, we obtain the theorem given (Art. 141), that every quantic of odd degree has a quadratic covariant. 159. Concomitants of quantics in three or more variables are expressed in a manner similar to that already explained. If 144 SYMBOLICAL REPRESENTATION OF CONCOMITANTS. ^iyi^i> ^JUJ^iy ^iHzh'i ^ cogredient sets of variables, then, by the rale for multiplication of determinants, the determinant is an invariant, which, by ti'ansformation, becomes a similar function multiplied by the modulus of transformation. And if in d d the above we write for cc^, -r^ ; for ^^, -^ } and so on, we obtain an invariantive symbol of operation, which we shall write 123. When, then, we wish to obtain invariants or covariants of any function U^ we have only to opel*ate on the product U^U^U^...Uf with the product of any number of symbols 123' 124^ 235^ &c, and after differentiation suppress all the suffixes. Thus, for example, let Z7^, £^, U^ be ternary quadricS| and let the coefficients in U^ be a, i, c, 2/^, 2^, 2A, as at p. 99, then 123' expanded is 4 2/((7'r + fK - a'/"- a'V) + 2g {Jif^ r/'- jy'-i V) + 2A(/y'+/Y-o'r-c'T); and this when we suppose the three quantics D^, IT, JJ^y to be identical, or a = a' = a'' &c. reduces to six times ale + 2/^A - af - Ig^ - cV. If in the above we replace a, the coefficient of a;*, by ^-g^ &c. we get the expansion of 123' as applied to any ternary quantic This covariant is called the Hessian of the quantic. It is seen, as at Art. 153, that odd powers of the symbol l23 vanish when it is applied to a single quantic. We give as a further example the expansion of 123^ applied to the quartiC| + 6 ((/y V + e«V -\-fxY) + ^^xyz {lx + my + nz). Then 123* is ahc - 4 (a J3C, 4 hc^a^ 4 cajb^) 4 3 [ad^ 4 5e' 4 cf*) 4 4 (a.i.Cj 4 ajkfi;^ — 1 2 {a^nd 4 %rnd 4 J^we 4 J,& 4 Cjmf+ cjtf) 4 12 [Ib^c^ 4 W2C,a, 4 wa.S,) 4 12 (rfP 4 m' 4/n') 4 6&/- 12ZMfi. IN ANY NUMBER OF VARIABLES. 145 160. We can express in tbe same manner fiinctions containing contragredlent variables ; for if a, yS, 7 be any variables contra- gredient to a?, y, z^ and therefore cogredient^with ^i 'j~ i T % it follows, as before, that the determinant ^ ^\dy^dz^ dy^dzj \dz^dx^ dz^dxj \dx^dy^ dx^dyj (which we shall denote by the abbreviation al2) is an inva- rlantive symbol of operation. Thus, If Z7„ C^ be two diflferent qnadrlcs, a]2'' is the contravariant called ^ [Conies^ Art. 377), which expanded is a«(iV'+iV- 2/y'O +/3'(cV'+cV- 2gY)-^rf (aT'+a"6'-2A'r) + 2/37 (/r+ fK- dr^ aT) + 27a (*'/"+ AT- jy- by) + 2a/3 [fY +/ Y - dK' - c'T), and which, when the two quadrlcs are identical, becomes the equation of the polar reciprocal of the quadrlc. In like manner, the quantic contravariant to a quartic, which I have called 8 {Higher Plane Curves^ p. 78), may be written symbolically al2^, and the quantic T in the same place may be written al2* a23' a31*. In any of these we have only to replace the coeflGicient of any power of a;, a;* by -7-» &c. to obtain a symbol which will yield a mixed concomitant when applied to a quantic of higher dimensions. Thus al2' is Ad^Ud'U /d'U\') J, ^ W "5?" l^j ;+*"•» which, when applied to a quadrlc. Is a contravariant, but, when applied to a quantic of higher order, contains both Xj y, 0, as well as the contragredlent a, )3, 7, and, therefore, is a mixed concomitant. In general. If we have the symbolical expression for any invariant of a binary quantic, we have only to prefix a con- travariant symbol a to every term, when we shall have a contravariant of a ternary quantic of the same order. And in particular it can be proved that If we take the symbolical ex- pression for the discriminant of a binary quantic, and prefix la U 146 SYMBOLICAL REPRESENTATION OF CONCOMITANTS. this manner a contravariant symbol to each term, we shall have the expression for the polar reciprocal of a ternary quantic. Thus, the symbol for the discriminant of a binary cubic is 12'.34*.13-24, and the polar reciprocal of a ternary cubic is al2Sa34'«al3.a24, which is obviously of the sixth order in the variables a, iS, 7, and of the fourth in the coefficients. fi a a 161. If in any contravariant we substitute j~ 1 j" 1 7* foi^ a, ^, 7, and operate on U, we get a covariant (Art. 1 39) ; and the symbol for this covariant is got from that for the contra- variant by writing a new figure instead of a. Thus from a23* is got 123^ from a23.a24 is got 123.124, &c. Conversely, if in the symbol for any invariant we replace any figure by a contravariant symbol a, we get the evectant of that invariant Thus, 123.124.234.314 is an invariant of a cubic, and the evectant of that invariant is 123.al2.a23.a31. In the case of a binary quantic, this rule assumes a simpler form ; for if we substitute a contravariant symbol for 1 in 12, it becomes, when written at full length, f -7 — ^ ;7" » ^^* since ^ and 7) are cogredient with — y and a;, this may be written ^ j"^y j" ) s^d ^^7 b® suppressed altogether, since it only affects the result with a numerical multiplier. Hence, given the symbol for any invariant of a binary quantic, its evectant is got by omitting all the factors which contain any one figure. Thus, 12".34M3.24 being the discriminant of a cubic, its evectant, got by omitting the factors which contain 4, is 12^.13* -rn» • i* . 1 • OU dU dU Ifm a contravariant ofany quantic we substitute ^T-, -y— , -r— (tie dy uz for a, )3, 7, we also get a covariant, and the symbol for it is obtained from that for the contravariant by writing a different MODIFIED NOTATION. 147 new figure In place of every a. Thus, from ail' we get 134.234; and so on. 162. In the explanation of symbolical methods which lias been hitherto given, I have followed the notation and course of proceeding originally made use of by Prof. Cayley. I wish now to explain some modifications of notation introduced by Aronhold and Clebsch, who have employed these symbolical methods with great success, but who perhaps at first scarcely sufficiently recognized the substantial identity of their methods with those previously given by Prof. Cayley. The variables are denoted a;,, a;,, a;,, &c., while the coefficients are denoted by suffixes corresponding to the variables which they multiply. Thus the ternary cubic, the ternary quartic, &c., may be briefly denoted Sia^j^^jfiCf^ ^^tum'^i^k^t^mi ^^*9 ^^^i^ the numbers i, ij Z, m are to receive in succession all the values I9 2, 3, &c. It will be observed that in this notation a^ = a^^^ = a^^^, so that when we form the sums indicated we obtain a quantic written with the numerical coefficients of the binomial theorem. Thus when we form the sum Sa,-^/r,a;^a?^, the three terms CL^^jx^x^i^ti ^i8t^i*^8^i9 ^sii^s^i^i ^® identical, as in like manner are the six terms ^l»8^I*2®») ^1W*l®8^2» ^218^8^1^3) ^28t^8^8^t) ^818^8^l^8» ^88l^8*8^ll 80 that the sum written at length would be «lll«^A^l •+ «888^8«8^8 •+ ^888^8^8^8 + ^<^U%^l^X^% +— + 6«188^1«2^8« And so. In like manner, In general. Now Aronhold uses, as an abbreviated expression for the quantic in general, where, after expansion, we are to substitute for the products afl^a^^ &c., the coefficients a,-^^. Thus the ternary cubic given above may be written in the abbreviated form the terms aflfl^x^x^x^ + Zafifl^x^x^x^ + &c in the expansion of the cube being replaced by a,,,a;,a;^a;„ 3«ii8^i^i^2) ^^* ^^^ quantity a^x^-V ajo^-\- a^x^ is written a^ 148 SYMBOLICAL REPRESENTATION OF CONCOMITANTS. or sometimes simply a, and the quantic is symbolically ex- pressed as a/. The cubic might equally have been written (&,«, + J^ajg + JgOjg)', (CjiCj + CgiTj + CjiTj)', &c., it being understood that we are in like manner to substitute for bfifi^^ ^i^Ai &c«j ^^e coeflGicients a^^^^ a„„ &c. Now the rule given by Aronhold for the formation of invariants is to take a number of deter- minants, whose constituents are the symbols a„ a,, a^ ; &„ b^^ &c.| to multiply all together, and after multiplication to substitute for the symbols a/z^ajj hj}jb^j the coefficients a,-,;, a^npj &c. Thus Aronhold first discovered a fundamental invariant of a ternary cubic by forming the four determinants S ± ctfi^c^^ SiijCjrfj, S±c,(fgag, l^±dfljb^] multiplying all together and then performing the substitutions already indicated. This is the same invariant which, in Prof. Cayley's notation, would be designated as 123.234.341.412. In order to obtain an in- variant by this method, It is obviously necessary (as in Art. 154) that the a symbols, b symbols, &c. respectively should each occur n times. A product of determinants not fulfilling this condition is made to express a covariant by joining to it such powers of a^, J^, &c. as will make up the total number of a's, b\ &c. to n. Thus the Hessian of a binary quadratic, which in Cayley's notation is 12* is in Aronhold's (aJ/; but the Hessian of any other binary quantic, which in Cayley's notation is still 12", is in Aronhold's (a5)'a/"*J^ 163. In order to see the substantial identity of the two methods, it is sufficient to observe that by the theorem of homo- geneous functions any quantic u of the n** order differs only by a numerical multiplier from {^i j~+^«t~ +^8;7~') ^) so that If we write it {aiOi;^ + a^x^-\'a^xJ'y the symbols a^y a^, a, differ only by a numerical constant from the differential sym- bols ~r- , &c. And we evidently get the same results whether with Prof. Cayley we form determlnkiits whose constituents are ^— , -7— , ^— , or with Aronhold^ whose constituents are ^1) «aJ ^»- MODIFIED NOTATION. 149 And the artifice made use of by both is the same. If we multiply together a number of differential symbols [-y--F "^ tA i'T"^ M'JTi 1 ^^'i *^^ operate on Uy it is evident the result will be a linear function of differentials of U of an order equal to the number of factors multiplied together ; and that in this way we can never get any power higher than the first of any differential coefficient. When, then, it is required to express symbolically . a function involving powers of the differential coefficients, the artifice used by Prof. Cayley was to write the function first with different sets of variables, and form such a function as (3^ + ^ j^) i'l — ^^77") ^»^' *^^ after differentiation to make the variables identical. So in like manner Aronhold in his symbolic multiplication uses different symbols which have the same meaning after the multipli- cation has been performed. By multiplying together symbols a,., a^, a^ &c., we can only get a term such as a^^j of no higher than the first order in the coefficients. When, then, we want to express symbolically functions of the coefficients of higher order than the first, the artifice is used of multiplying together different sets of symbols a,., a^, a^; 5^, 5^, i^, &c., the products ^P'lP'D ^i^A) ^fiif^n &C'> ^ equally denoting the coefficient a,-^^. The notations explained in this Lesson afford a complete calculus, by means of which invariants and covariants can be transformed and the identity of different expressions ascer- tained. We shall in a subsequent Lesson give illustrations of the applications of this method, referring those desirous of further information to Clebsch's valuable Theorie der binaren algebraischen Formerly in which work this symbolical method is the basis of the whole treatment of the subject. ( 150 ) LESSON XV. CANONICAL FORMS. 164. Since invariants and covariants retain their relations to each other, no matter how the qaantic is linearly transformed, it is plain that when we wish to study these relations it is suffi- cient to do so by discussing the quantic in the simplest form to which it is possible to reduce it. This is only extending to quantics in general what the reader is familiar with in the case of ternary and quaternary quantics; since, when we wish to study the properties of a curve or surface, every geometer is familiar with the advantage of choosing such axes as shall reduce the equation of this curve or surface to its simplest form.* The simplest form then, to which a quantic can without loss of generality be reduced, is called the canonical form of the quantic. We can, by merely counting the constants^ ascertain whether any proposed simple form is sufficiently general to be taken as the canonical form of a quantic, for if the proposed form does not, either explicitly or implicitly, contain as many con- stants as the given quantic in its most general form, it will not be possible always to reduce the general to the proposed form.f * It mnst be owned, howeyer, that as in the progress of analysis greater facility ia gained in dealing with quantics in their most general form, the advantage diimnishes of reducing them to simpler forms. f It is not true, however, conversely, that a form which contains the proper number of constants is necessarily one to which the general equation may be reduced. For when we endeavour by comparison of coefficients to identify such a form with the general equation, although the number of equations is equal to the number of quantities to be determined, it mat/ happen that the constants enter into the equatioiui in such a way that all the equations cannot be satisfied. Thus {x - o)2 + (y - /3)2 = & + my + » is a form containing five constants, and yet is not one to which the general equation of a ternary quadric can be reduced ; since the constants enter the equation in such a way that though we have more than enough to make the coefficients of x and y and the absolute term identical with those in any proposed equation, we have no meaDB of identifying the coefficients of x^f xy and y^, A more important example is a^ + y* + s* + «*-♦-»«, OF QUADRICS. 151 Thus, a binary cubic may be reduced to the form X' -f F' ; for the latter form, being equivalent to [hs + myY + {I'x + w'y)', con- tains implicitly four constants, and therefore is as general as (a, J, c, dyxj yY. So, in like manner, a ternary cubic In its most general form contains ten constants; but the form X' + F' + Z' + 6?wXyZ contains also ten constants, since, in addition to the m which appears explicitly, X, F, Z implicitly involve three constants each. This latter, then, may be taken as the canonical form of a ternary cubic, and, in fact, some of the most important advances that have been made in the theory of curves of the third degree are owing to the use of the equation in this simple and manageable form. 165. The quadratic function (a, 5, cX«, yY can be reduced in an infinity of ways to the form 05" + y", since the latter form implicitly contains four constants, and the former only three. In like manner the ternary quadrie which contains six constants can be reduced in an infinity of ways to the form iXi^-^y^ + ^i since this last contains implicitly nine c(Histants; and, in general, a quadratic form in any number of variables can be reduced in an infinity of ways to a sum of squares. It is worth observing, however, that though a quadratic form can be reduced in an infinity of ways to a sum of squares, yet the number of positive and negative squares in this sum is fixed. Thus, if a binary quadrie can be reduced to the form a?' + y', it cannot also be reduced to the form u* — v*, since we cannot have a5* + y* identical with w* — v*, the factors on the one side of the identity being imaginary, and those on the other being real. In like manner, for ternary quadrics we cannot have a?* + y* — 2' = m" + v* -f ic?*, since we should thus have a* + y* = «' + tt* + V* + !£?', or, in other words, and if we make x and y = 0, one side of the identity would 1 ■ I I ■ - I where z,u,v are linear functions. In the case of a ternary quantic this form contains implicitly fourteen independent constants, and therefore seems to be one to which the quartic in general can be reduced. But Clebsch has shewn that a condition must be fulfilled in order that a quartic should be reducible to this form, namely, the Tazuflhing of a certain iuTanant. See also Swrfacei, Note to Art. 235. 152 CANONICAL FORMS. vanish, and the other would reduce itself to the sum of four positive squares which could not be = 0. And the same argu- ment applies in general. 166. We commence by shewing that, as has been just stated, a cubic may always be reduced to the sum of two cubes. To do this is, in fact, to solve the equation, since when the quantic is brought to the form X^ + F®, it can immediately be resolved into its linear factors. Now, if the cubic (a, J, c, d\x^ yf become by transformation (-4, B^ C, D\X^ F)', then, since (Art. 126) the Hessian (oo: + Jy) (ca: + e?y) — ( Jaj + cy)" is a co- variant, it will, by the definition of a covariant, be transformed into a similar function of A^ B^ C, i>, X, Y. That is to say, we must have {ac - 6") a* + [ad- Ic) xy + [Id- c") y* = [A G- B') X» + (AD - BC) Xr+ [BD-C) Y\ Now, if in the transformed cubic, B and C vanish, the Hessian takes the form ADXY] and we see at once that we are to take for X and Y the two factors into which the Hessian may be broken up. "When we have found X and F, we compare the given cubic with -4Jf*-f 2>F*, and determine A and D by comparison of coefficients. Ex. To reduce 4a!« + 9a* + 18a; + 17 to the form Ay* + i)r». The Hessian is (4aj + 8) (6a; + 17) - (3a; + 6)», or 16a;* + 60a; + 16, whose linear factors are a; + 3, d« + 1. Gompaiing then the given cubic with ^ (a; + 3)» + D (3a; + 1)», we have ^ + 27i) = 4, 27A + D- 17, whence 728J9 = 91, 728.1 = .465, or -4 is to D in the ratio of 5 to 1. The given cubic then only differs by a factor (viz. 8) from 6(aj+3)»+(3a;+l)», and it is obvious that the roots of the cubic are given by the equation 3a;+l + (a; + 3)8^(6) = 0. 167. It is evident that every cubic cannot be brought by reaZ transformation to the form JX' + -DF*, for this last form has one real factor and two imaginary; and therefore cannot OP QU ANTICS OF ODD DEaREE; 153 Ibe identical with a cubic whose three factors ajris i^eal. The discriminant of the Hessian 4(ac-J»)(JtZ-c«)-(arf-Jc)' is, with sigii changed, the same as that of the ctibic. When the discriminant of the cubic is positive, the Hessian has two real factors, and the cubic one real factor and two imaginary. When it is negative, the Hessian has two imaginary factors, and the cubic three real. When it vanishes, both Hessian and cubic have two equal factors, and it can be directly verified that the Hessian of X^Y is X*.* It is to be observed, that a quantic of the same degree cannot always be reduced to the same canonical form. The impossibility of the reduction indicates some singularity in the form of the quantic. Thus a cubic having a square factor cannot be brought to the form Ax^-^- Dy^ : a different canonical form must be adopted, and the most simple one is the form a?y^ to which the cubic in question is obviously at once reducible. 168. In the same manner as a cubic can be brought to the sum of two cubes, so in general any binary quantic of odd degree (2n — 1) can be reduced to the sum of n powers of the (2n — 1)** degree, a theorem due to Dr. Sylvester. For the number of constants in any binary quantic is always one more than its degree, or, in the present case, 2n ; and we have the same number of constants if we take n terms of the form (faj + ?wy)*""*. The actual transformation is performed by a method which is the generalization of that employed (Art. 166). For simplicity, we only apply it to the fifth degree, but the method is general. The problem then is to determine t/, t?, w^ so that (a, J, c, c?, e,/][a;, yY may = m* + 1?* •+ w^. Now we say that if we form the determinant ax + hy^ bx + cy^ cx-)r dy bx -V cyj CX + dyj dx'{-ey cx-i-dyj dx + eyj ex +/y * In general, when a binary quantic has a square factor, this will also be a square factor in its Hessian, as may be verified at once by forming the Hessian of a^<f>. 154 CANONICAL FORMS. the three factors of this cubic ^ill be w, v, w. For let then, differentiating the identity (a, 5, c, c?, e,/3[a7, y)' = m* 4 w* + 1^*^ four times successively with regard to Xj and dividing by 120, we get ax + bt/ = Vu + V% + r*w?. Similarly differentiating three times with regard to x^ and once with regard to y, lx-\-cy- Vmu + Pm'v + r^'/w^i^? ; and so on. The determinant, then, written above, may be pnt Into the form Pw + l"v + r*w , rmu-{-rm'v+rm"w, ?mV+ZVt;+rm"»tt? Pmu+rm'v+rm"w, rm'u-^Wv-^rWw, lm''u+rm"vfrm"'w But (Art. 22) this is the product Pu, Pv, rw ?, p, r Imuj Tm'v, l"m"w Im, I'm', Vni jw", m'*, »»'" ./' or is • When, then, the determinant written in the beginning of this Article has been found, by solving a cubic equation, to be the product of the factors (aJ + Xy) {a; + /A^) (aj+ f^), we know that t«, V, w can only differ from these by numerical coefficients, and we may put (a, J, c, d, e,fJx,yY = A[x-^\yY^'B[x'\-iiyY'{'G[x + vyY^, and then -4, 5, C are found by solving any of the systems of simple equations got by equating three coefficients on both sides of the above identity. The determinant used in this Article Is a covariant, which is called the canonizanf of the given quintlc. CANONIZANT OF QUINTIC. 155 169. The canonizant may be written in another, and perhaps simpler form, namely, a, h Cj d h c, d. e c, d, «. f This last is the form in which we should have been led to it if we had followed the course that naturally presented itself, and sought directly to determine the six quantities -4, B^ (7, \, /i, k, by solving the six equations got on comparison of coefficients of the identity last written in Art. 168. For the development of the solution in this form, to which we cannot afford the necessary space here, we refer to Sylvester's Paper [Phtlo^ sophical Magazine^ November, 1851). Meanwhile, the identity of the determinant in this Article with that in the last has been shown by Prof. Cayley as follows. We have, by multiplicatiou of determinants (Art. 22). }ti - y'^) »«'> - »' a. y\ d, d e f 1, 0, 0, X, y, 0, 0, X, y, 0, 0, a?, y 0, ax + hy^ bx + cy, ca? + dy 0, bx + cyj cx+dyj dx-\- ey 0, ox-k-dy^ dx + ey^ ex+fy which, dividing both sides of the equation by y', gives the identity required. 170. We have still to mention another way of forming the canonizant. Let this sought covariant be (-4, J?, <7, IfJ[xj y)% where we want to determine -4, -B, G^ D\ then (Art, 140) (-4, B^ (7, Jy%-r J — -T-y will also yield a covariant. But if this operation is applied to (i» + Xy)'* where x-^\y is a factor in (-4, B^ Gy I)\xy yfy the result must vanish, since one of the 156 CANONICAL FORMS. ft fl factors m the operatipg symbol is ^ — ^ 1' * Since, then, the giTen quantic is by hypothesis the sum of three terms of the form (qj + Xy)*, the result of applying to the given quantic the operatipg symbol just written must yanish. Thus, then, we have A (c?, e,fjx^ yf - P (c, d, e$x, jtY + (S, c, d\x, yY - £> (a, by cXxj yY = 0, or, ec[uatiqg separately to the coefi|cieuts Qf a;', q^^ y% we have 4d-Bc-¥Cb"Pa=^0j 4f-Be-^Cd^Dc=^% whence (Art. 28} A is proportional to the determinant got by a, by c suppressing the column A or ^nd SQ for jj?, C7, i>. by Cy d Cy dy e which yalues give for the caponi^ant the form stated in tbe last Article. 171. "We proceed now to quantics of even degree (2n). Since this quantic contains 2n + 1 terms, if we equate it to a sum of n powers of the degree 2n, we haye one equation more to satisfy than we haye constants at our disposal. On the other hand, if we add another 2n^ power, we have ope constant too many, and the quantic can be reduced to this form ip an infinity of ways. It is easy, hqweyer, to determine the pondition that the given quantic should be reducible to the sum of ti, 2n^ powers. Thus, for example, the conditions that a quartic should be reducible to the sum of two fourth powers, and that a sextic should be reducible to the sum of three sixth powers, are respectively the determinants a, by Cy d ay by c . by Cy d = 0, Cy dy e by Cy dy C Cy dy Cy f = 0, OF QUARTIC. 157 and so on. For, in the case of the quartic, the constituents of the determinant are the several fourth differentials of the quantic, and expressing these in terms of u and v precisely as in Art. 168, it is easy to see, Art. 26, Ex. 5, that the determinant must vanish, when the quartic can be reduced to the form t«^ + t;\ Similarly for the rest. This determinant expanded in the case of the quartic is the invariant already noticed (see Art. 141, Ex. 1), ace + 2hcd—ad* - eV — c*. 172. When this condition is not fulfilled, the quantic is re- duced to the sum of n powers, together with an additional term. Thus, the canonical form for a quartic is naturally taken to be w* + 1?* + 6XmV. We shall commence with the reduction of the general quartic to this canonical form; the method which we shall use is not the easiest for this case, but is that which shows most readily how the reduction is to be effected in general. Let the product, then, of u^ t;, which we seek to determine, be (-4, -B, C\x^ y)*, and let us operate with (-4, -B, C^-t- , — ^r)* on both sides of the identity (a, J, c, J, ejaj, y)* = m* + v* + 6Xm V. Now, as before, this operation performed on u^ and on v^ will vanish, and when performed on 6XuV it will be found to give 12Vmv, where X' = 2 (4^(7— 5'*) X. Equating then the coefficients of x^^ xy^ and t^ on both sides, after performing the operation, we get the three equations Ae^Bd^-Cc^ X'a, whence eliminating A^ B^ (7, we have to determine X^, the determinant a, J, c — X' J, c + ^X', d c — X', rf, e = 0, 158 CANONICAL FORMS. 'which, expanded, is the cubic the coefficients of which are invariants. Thus, then, we have a striking difference in th^ reduction of binary quantics to their canonical form, between the cases where the degree is odd and where it is even. In the former case, the redaction is unique, and the system w, v^ m?, &c. can be determined in but one way. When u is of even degree, however, more systems than one cau be found to solve the problem. Thus, in the present instance, a quartic can be reduced in three ways to the canonical form, and if we take for V any of the roots of the above cubic, its value substituted in the preceding system of equations enables us to determine -4, Bj G. 173. If now we proceed to the investigation of the reduction of the quantic (a^, a„ a^ ...Xit*, y)*", the most natural canonical form to assume would be u^ + u''" 4- w*" + &c. + XuVici* &c., there being n quantities u^ v^ w^ &c. But the actual reduction to this form is attended with difficulties which have not been overcome, except for the cases w = 2 and w = 4. But the method used in the last Article can be applied if we take for the canonical form u^ + «?'*" -f &c. -f \ Vuvw &c., where, if uvw &c. = (^^, ^j, A^ ...Jx^ y)", F is a covariant of this latter function such that when Vuvw &c. is operated on by {A^^ A^ •••5Cx ) "" x)*) ^^® result is propor- tional to the product uvw &c. Suppose, for the moment, that we had found a function V to fulfil this condition, then, pro- ceeding exactly as in the last Article, and operating with the differential symbol last written on the identity got by equating the quantic to its canonical form, we get the system of equations 2 • ■ J II L * N.B. — ^The discriminant of this cubic is the same as that of the qnartic OF QUANTICS OF EVEN DEGREE. 159 whence, eliminating A^^ A^^ A^^ &c., we get the determinant ^«-^') ««-,) a W-2J a n+l) «« + -^'j Vl» a. a. ^»i+a» ^«+i) 1.2 ^« "~ — 'i 7\ ^1 ••• d^ a Sn9 « Sa-i9 a M~29 ff T V* and having found V by equating to this determinant expanded (a remarkable equation, all the coefficients of which will be invariants), the equations last written enable us to determine the values of -4^, -4^, &c., corresponding to any of the w + 1 values of X'. 174. To apply this to the case of the sextic, the canonical form here is u® + v" + w* + Vuvw^ where, if uvw be V is the evectant of the discriminant of this last quantic, and its value is written at full length (Art. 142). Now it will afford an excellent example of the use of canonical forms if we show that in any cubic the result of the operation performed on the product of the cubic and the evectant just mentioned, will be proportional to the cubic itself. For it is sufficient to prove this, for the case when the cubic is reduced to the canonical form aJ^^-y^ in which case the evectant will be x^ - y', as appears at once by putting J = c = 0, and a = d=l in the value given, Art. 142. The product, then, of cubic and * The determinant above written may be otherwise obtained as follows. Leb sc^y y' be cogredient to Xj y, and let ns form the function which (Arts. 125, 131) we have proved to be linearly transformed into a function of similar form. Equate to zero the n + 1 coefficients of the several powers ar", x*-^yy <fec.y and from these eliminate linearly the n + 1 quantities a;'", x'^-^y', &c., and we obtain the determinant in question. 160 CANONICAL FORMS. evectant will be a:*— y*, which, if operated on by ^ "*• ;r s ) gl^^s a result manifestly proportional to sc' + y*. And the theorem now proved being independent of linear transformatioii, if true for any form of the cubic, is true in general. The canonical form, then, being assumed as above, we proceed exactly as in the last Article, and We solve for X from the equation «0» «« «.i «3- -X «» «.i «.+ i\ «4 «,» «»- ■i's «4> «i a, + \, «« «». «« ^0, which, when expanded, will be found to contain only even powers of \. If we suppose uvw reduced as above to its canonical form a?' + y, the three factors of which are where o> is a cube root of unity, then it is evident from the above that the corresponding canonical form for the sextic is ^ (a: + y)* + -B (a; + ©y)* + (7 (a; + ©"y)* + D (»• - /). It can be proved that if t/, 2;, u? be the factors of the cubic, then the factors of the evectant used above are v — w^ w — u^ u- Vj so that the canonical form of the sextic may also be written u^-^v'^ + w^ + Xuvw {u — v) (v — w) [to — w). 175. In the case of the octavic the canonical form is for if we operate on u^v'wW with a symbol formed according to the same method as in the preceding Articles, the result will be proportional to uvwz. As for higher canonical forms we content ourselves with again mentioning that for a ternary cubic, viz. x^ + y^ + z^ + 6mxyz^ and that given by Sylvester for a quaternary cubic, ( 161 ) LESSON XVL StSTHMS OF QUANTICS. 176. It still remains to explain a few properties of systems of qaantics, to which we devote this Lesson. An invariant of a system of quantics of the same degree is called a cambinant if it is unaltered (eitcept bj a constant multiplier) not onlj when the variables are linearly transformed, but also when for any of the quantics is substituted a linear function of the quantics. Thus the eliminant of a system of quantics u^ v^ w is 9, conH binant. Forr, evidently the result of substituting the common roots of vw in u + Xv + fiw is the same as that of substituting them in u ; and the eliminant of t« + Xt; + fi^j v^ w is the same as the eliminant of urno. In addition to the differential equa- tions satisfied by ordinary mvariants, combinants must evidently also satisfy the equation a'dl Vdl c'dl - -^ + --^ + --T- + &c. = 0. aa do dc It follows from this that in the case of two quantics a combinant is a fiinction of the determinants [aV)y [ac')^ ^^)^ &c*; ii^ the case of three, of the determinants {aVc\ &c. ; and will accord* ingly vanbh identically, if any two of the quantics become identical. If we substitute for w, v ; Xw + /av, Wu + fiv^ every one of the determinants [aV) will be multiplied by (X/a' — X'/a) J and therefore the combinant will be multiplied by a power of (X/A^ - XV) equal to the order of the combinant in the co- efficients of any of the quantics. Similarly for any number of quantics. There may be in like manner combinantive covariants, which are equally covariants when for any of the quantics is substituted a linear function of them. For instance,^ the Jacobian (Art. 88) «,) «,) «. 162 SYSTEMS OF QUANTICS. if we substitate for u, lu + wv + nwj for v, Vu + rr/v + w'tt?, &c. by the property of determinants, becomes the product of the determinants (?wV'), {u^v^tv^). The coefficients of a combinan- tive covariant are also functions of the determinants (oi^), (ac^) ; (oJV), &c. 177. If tt = (a, i, c.Ja?, y)*, r = (o', b% c'-.-Jxc, y)" be any two binary quantics of the same degree, then u-\-kv or (o+Aa', 6+ AA',..3[a;, y)", where we give different values to A, denotes a system of quantics which are said to form with u, v an involution. Now there will be in general 2(n— 1] quantics .of the system, each of which will have a square factor. For 'the discriminant of a quantic of the n** degree is of the order 2(w— 1) in the coefficients (Art. 105). If then we sub- stitute a + Jea' for a, h f W for J, &c., there will evidently be 2 (n — 1) values of A;, for which the discriminant will vanish. If we make y = 1 in any of the quantics, it denotes n points on the axis of a?. We have just proved that in 2(w— 1) cases, two of the n points denoted by u-\-kv will coincide; or, in other words we may say, that there are 2(w— 1) double points in the involution. When u + kv has a square factor a? — a, we know that a satisfies the two equations got by differentiation, viz. m, + kv^ = 0, Wj + iVj = 0, and therefore will satisfy the equation got by eliminating k between them, viz. m,v, — u^v, = 0. Now MjV, — WjV^, which is of the degree 2(n— I), is the Jacobian of Uj V ; and we see that by equating the Jacobian to 0, we obtain the 2 (n - 1) double points of the involution determined by 178. If u and v have a common factor^ this will appear as a square factor in their Jacobian. First, let it be observed, that since wm = a;w^ -f yw^, nv = xv^-{-yv^^ then if we write J for u^v^ — Wj,v„ we shall have n {uv^ - vu^ = xJ^ n {uv^ - vm J = — yj. * In like manner, for a ternary quantic, the Jacobian of u, r, to is the loons of the double points of all curves of the Bystem u + kv + Ito which have double points. And similarly for quantics with 'any number of .variables. INVOLUTION. J63 Differqntiating the first of these equations with reg^d to y^ anil the second with regard to a?, we get n [uv^ - vuj = xJ^, n (wv,, - t?w„) = - yJ^. It follows from the equations we have written, that any value a of X which makes both u and v vanish, will make not only J vanish but also its differentials e/^, J,, and therefore a: — a must be a square factor in J. Or more directly thus: let m=^^, v^lSyjt^ where I3=lx+fnyi then Wj= ?^ + ^80^, u^=m<f> + I3<f>^j v^-l'^ + fiyfr^^ t;,= m^ + )8^^ j and u^v^- w,v =^{0^^,- ^8^,)-i- ^^(^^,-^»^)+^w(^^i(r-^^J^ whence (n - 1) (w^v, — w,v,) = (n - 1) ^- (0,^. - ^.^J + /S (tr + ^y) (^,^. - ^.V^,) It follows from what has been said, that the discriminant of the Jacobian of u, v must contain B their resultant as a factor ; since whenever B vanishes, the Jacobian has two equal roots. Thus in the case of two quadratics. (a, J, cXx, y)% {a\ b\ cjx, y)% the Jacobian is [aV) x^ + {ad) xy + (Jc') y*, whose discriminant is 4 [aV) {be') - [ae')*^ which is the eliminant of the two quadratics. In the case of quantics of higher order, the discriminant of the Jacobian will, in addition to the resultant^ contain another factor, the nature of which will appear from the following articles. 179. It has been said that we can always determine A;, sa that u-\-kv shall have a square factor. But since two conditions must be fulfilled, in order that u-^-kv may have a cube factor, h cannot be determined so that this shall be the case unless a certain relation connect the coefiicients of u and v. This condi" tton will be of the order 3 (n - 2) in the coefficients both of u and v. If {x — a)' be a factor in u + lev •\- Iw^x-^ a will be a factoi: in the three second differential coefficients, or a; = a will satisfy the equations SYSTEMS OF QUANTICS. whence eliminating k and l,x=a will satiafy the equation If then we nse the word treble-point in a aense analogous to that in which we used the word double-point (Art. 177), we aee that the equation which has been juat written gives the treble points of the eystem u + kv + lio; and aince the equation la of the degree S{n — 2), there may be 3 [n — 2) such treble points. But we could find the number of treble polntfl otherwise. Suppose we have formed the condition that u + kv should admit of a treble point, and that this condition ia of the order p in the coefficients of m. If in thia condition we sub- stitute for each coefficient [a) of u, a + la", we get an equation of the degree p in I; and therefore p values of I will be found to satisfy it. In other words, p quanlica of the Bystem u + kv+lw will have a treble point. It follows then from what haa juflt been proved that p = 3[n — 2). And the same argu- jaent proves that the condition in question is of the order 3 (n — 2) in the coefficients of v. This condition is evidently a eombinant ; for if it is possible to give such a value to k, as that u + kv shall have a cube factor, it must be possible to determine k, so that (w + mv) + h> Bball have a cube factor. 180, If u + kv have a cube factor {x — af, then the Jacobian of u and v will contain the square factor (a; — a)". For the two differentials m, + Ai;,, u^ + kv^ wiU evidently contain this square factor, and therefore it will appear also in the Jacobian, which may be written (M[ + hv,) r, — (w, + kv^) v,. If then S = be the condition that u + kv may have a cube factor, S will be a factor in the discriminant of the Jacobian, since if S= the Jacobiao baa two equal roots, and therefore its discriminant vanishes. If ,R be the resultant, the discriminant of the Jacobian can only differ by a numerical factor from ^8. For since the Jacobian is of the degree 2[ii — I], its discriminant is of the degree 2 {2 (w - 1) - 1} in its coefficients, which are of the first order in the coefficients of both u and v. How ^ ia of the order INVOLUTION. 165 ft in each set of coefficients, 8 of the order 3 (n — 2). Both these are factors in the discriminant ; and it can have no other, since n + 3(«-2)=2{2(n-l)-l}. 181. The discriminant ot u + kv^ considered as a function of kj will have . a square factor whenever u and v have a common factor. In fact (Art. Ill) the discriminant of u-^kv will be of the form (a + ka) ^ -f (ft + Jcb'f •^. But if u and v have a common factor, we can linearly transform u and v 80 that this factor shall be y, that is to say, so that both a and of shall vanish. The discriminant will therefore have the gqnare factor [b + kby ; and since the form of the discriminant is not affected by a linear transformation of the variables, it always has a square factor in the case supposed. It follows that if we form the discriminant of te + Art?, and then the discriminant of this again considered as a function of % the latter will contam as a factor It the resultant of u and v. For it has been proved that when jS^O, the function of k has two equal roots, and therefore its discriminant vanishes. For example, the discriminant of a quadratic ac — ft* becomes, by the substitution of a + ka' for a, &c., (ac - 6») + A; (ac' + ca' - 2ftft') + AM«'<5' - ft''), whose discriminant is 4 (ac - ft*) (aV - ft") -'{a(/'\- ca' - 2ftftO*. But this is a form in which, as was shewn by Boole, the resultant of the two quadratics (a, ft, c§x^ y)*, [a\ ft', c'Ja?, yf can be written (cf. Fx. 6, p. 24). This form, all the component parts of which are invariants, is sometimes more convenient than that given (Art. 178). In the case of quantics of higher order, the discriminant of the discriminant will have £ as a factor, but will have other factors besides. 182. If u have either a cube factor or two distinct square factors, the discriminant of t« + Ait; will be divisible by A;'- For if the discriminant of u be A, that of t« + Art; is \ 166 SYSTEMS OF QUA15TICS. Now when.K has a square factor A yanishes; and it appear^ from the expressions in Art. 114, that if either three roots of U are equal a = /8 = 7, or two distinct pairs be equal a = /8, 7 = S^ then all the differentials of A, --7- j &c., vanish ; and therefore the coefficient of k in the expression just given vanishes.' The discriminant therefore contains k* as a factor. It is evident hence that if u-¥av have a cube or two square factors, the discriminant of u-^kv will be divisible by {k — af ; since u-\- kv may be written w + at;+ (A — a)t;. If then, as before, 5=0 express the condition that the series u-]- kv may include one quantic having a cube factor; and if 2^=0 be the condition that it should include one having two square factors, both 3 and T will be factors in the discriminant with respect to k of the discriminant of u + kv. For we have just seen that the discriminant has a square factor if either 5=0 or T=0. We proved in the last Article that the discriminant has JS as a factor ; and, in fact, the discriminant will be, as Prof. Cayley has observed, BS* T^. I do not know whether there is any more rigid proof of this than that we see that there is no other case in which the discriminant of u + kv has a square factor ; that we find in the case of the third and fourth degrees that 8 and T enter in the form 5', T* ; and that we can thus account for the order in general. For the discriminant of u-k-kv is of the order 2(n— 1) in A, and the coefficients are of the, order 0, 1, ..., 2 (n — 1) in the coefficients of either quantic. The discriminant then with respect to k will be of the order 2(n- 1) (2/1 — 3) in the coefficients of either quantic. But R is of the order w, 8 of the order 3 (w — 2), and it will be proved in a subsequent lesson that T is of the order 2 (« — 2) (w - 3), and 2(w-l)(2n-3)=w + 9(n-2)+4(w-2)(w-3), 183. It was stated (Art. 176) that every combinant of m, v hecomes multiplied by a power of {Xfi — \'fi) when we sub- stitute \u + fiv^ \'u + fiv for M, V. It will be useful to prove otherwise that the eliminant of w, v has this property. First, let it be observed that if we hav^ any number of quantica ELIMINAMTS. 167 one of which Is the prodact of several others, Uj v^ ww*w*\ their resultant is the product of the resultants [uvw)^ [uvw*)^ (uW). For when we substitute the common roots of u, v in the last and multiply the results, we evidently get the product of the results of making the same substitution in w^ w\ uf\ Again, the resultant of u, v, kw is the resultant of u, v, w multiplied by A;"*** since the coefficients of w enter into the resultant in the degree wn. If now 5 (w, v) denote the resultant of m, v, which are supposed to be both of the same degree n, we have yTR (\w + /Ltv, X'm + /^'t?) = R (X/a'm + i».fi!v^ \'u + ytt't;)* = B [ [\ii' - X» M, Vm + tifv] = (X/i' - X»* B (tt, X'm + ii'v) = (X/ - X»* /a'" B (m, v), whence 5 (Xw + a*v, X'm + fiv) = (X/a' — X'/tt)" jB (u, w). By the same method it can be proved that the eliminant of Xm + /Ltv + vw^ X'm + fi'v + /t^, X'^M + /a'^v + Vw is (XfiV^)* times that of t«, V, 2^7, and so on. 184. If J7, Fbe functions of the orders m and n respectively in u, t;, which are themselves functions of a?, ^ of the order p, and if D be the result of eliminating u, v, between 27, F; then the result of eliminating a?, y between [7, F will be 2/ times the Twn** power of the resultant of m, v. For U may be re- solved into the factors u - aw, m — I3v^ &c., and F into u — ot'r, w — yS'v, &c. And, Art. 183, the resultant of [7, F will be the product of all the separate resultants u- av^u — a'v. But one of these is (a — a')" B (m, v). There are mn such resultants. When therefore we multiply all together, we get the wm** power of B{ujv) multiplied by the p** power of (a — a') (a — a'')j &c. But this last Is the eliminant of U^ V with respect to u^ v. 185. Similarly, let it be required to find the discriminant| with respect to a?, y, of Z7, where [7 is a function of u, v. First, let it be remarked (see Art. 110) that the discriminant of the product of two binary quantics i/, v is the product of the * The resultant of u+kvy v, being the same as the resultant of u, v, Art. 17C^ We next subtract fi times the second quantic from the first. 168 SYSTEMS OF QUANTIC9. dlscriminaiita of u aod i; multiplied by the sijuare of tbeir reaaltant. If tbeo U^lu — av) [u — ^v) &c., the discrimiaant of U will be tbe product of tbe diacriniiaants of u — oy, u — /9«, &c. by tbe square of the product of all the separate resultants a — aw, u — /Sy, But, as before, any of these will be (a - fi)' E {u, v). If then m be the degree of U considered as a function of w, w ; there will be ^m (m — I] separate resultants, and the square of the product of all will be (a- /S)"" (a -7)% &c. x ^-<"-" {u, «). But (a — 0y {a — 7)', &c. is the diacriminant of U considered as a function of «, v. If then we call this A, we have proved that the product of the squares of the separate resultants is A*fi"'"~". Let us now consider the product of the discriminants of u — aw, M - (9i', &D. ; tins is the result of eliminating 6 betweea the discriminant of u—0v, which is a quantic of tbe order' 2{p-l) in B and the quantic of the m" order got by sub- stituting u=Sv in U. Or this product has been otherwise repreaented by Dr. Sylvester, If a„, b^ be the coefficients of «f in u, V, then (Art. 108) the resultant of m — aw, m, — au, will be 0—06 times the discriminant of « — aw. But S{u,v)M{u-i v,)=}i{u- '— aupj = E { U—dVy u^v—uv,). Now (Art. 178) ^ (M,f — wi),) =,y<7' where J is «,«, — «,«„ and E{u~aVjy) isa, — oig. It appears thus that the discriminant of M— ttw differs only by a namorical factor from the resultant of (w-ay, J) divided by B{u, v). The product then of all tbe discriminants will be the — resultant of J and the product u — oti, u — ^r, &c,, in other words, the resultant of U, J — divided by the m" power of B{u, v). Thus we have Dr. Sylvester's result [Comptes Beitdiis, LViii., 1078) that the dtscriminant of C7" with respect to 31:1/ is A''i?(K, ii)"'"""^[C7, /). But it will be observed that the result expressed thus is not in its most reduced form since B{U,J} contains the factor E («, vj". 186. We have next to see what corresponds in the case of ternary and quaternary quantica to the theory just explained for systems of binary quantlcs. Let then u and v be two ternary quantics, and let us suppose that we have formed the ELIMINANTS. 169 discriminant of u + kv. Then (for certain relations between the quantics u^ v) this discrimiDant considered as a function of k will have a square factor. In the first place the discriminant will have a square factor, if the curves represented by u and c) touch each other. For we have seen (Art. 117) that if the equation of a curve be (zsT + bz^'^x + czl^'^y -f &c. = 0, its dis* criminant is of the form ad 4- V^ + hc^ 4- f^x- ^^^ discriminant then of M + A;y will be of the form (a -f ha) ^ + (i + W)^ ^ + &c* But if we take for the point xy^ a point common to u and i^^ both a and cl will vanish; and if we take the line y for the common tangent, both h and V vanish; and the discriminant will be of the form (c-f Arc')*^; and therefore will always have a square factor in the case supposed. 187. Again, the discriminant will have a square factor if u have either a cusp or two double points. The vanishing of the discriminant A of a ternary quantic gives the condition that w„ Wj, u^ shall have a common system of values. If, however^ u have either two double points, or a cusp, m^, w^, u^ will have two systems of common values, distinct or coincident, and therefore (Art. 103) not only will A vanish, but also its difi^erentials with respect to all the coefficients of u. The discriminant then of u+ A;t^ being in general A + A(--i h — jr- + ^c. J + &c. will in this case be divisible by A;'. And as in Art. 182, it will be divisible by (A — of if the curve u-\-av have either a cusp or two double points. Let then 5 = be the tact-invariant of u and v, that is to say, the condition that the two curves should touch; ;8'=0 the condition that in the system of curves u-vkv shall be included one having a cusp; and 7=0 the condition that there shall be included one having two double points. It has been proved that jB, 5, T are all factors in the discriminant of the discriminant of u + ^z', considered as a function of It. In fact this discriminant will be BS^I". For an investi- gation of the orders of jB, 5, T, when both curves are of the same degree, see Higher Plane Curves^ Art. 399. The tact-invariant B is of the order 3n (w- 1) in the coeffi- z 170 SYSTEMS OF QU ANTICS. cients, 8 of the order 12 (n - 1) {n - 2), and T of the order f («— 1) (n — 2)(3«* — 3n— 11) : the discriminant of u-\-kv in regard to a; : y is of the order 3 (n — 1)^ and the discriminant of this with regard to k of the order 3 (n - 1)* (3n* — 6n + 2), and we have identically 8(n-l)*(3n"-6n + 2) « 3n (n - 1) + 36 (n - 1) (n - 2) + 3 <n - 1) (w - 2) (3n» - 3n - 11), showing that the order of the discriminant is equal to that ofHST. 188. The theorem given, Art 110, for the discriminant of the product of two binary quantics cannot be extended to ternary quantics ; for the discriminant of the product of two will, in this case, vanish identically. In fact, the discriminant is the condition that a curve shall have a double point ; and a curve made up of two others has double points; namely, the intersections of the component corves. Or, without any geometrical considerations, the discriminant of uv is the condition that values of the variables can be foond to satisfy simultaneously the differentials ur, + rtf,, ur^-hru^ &c» But these will all be satisfied by any values whidi satisfy simultaneously u and r ; and such values can alwaj^ be found when there are more than two variables. But the theorem of Art. 110 mav directlv be extoided to tact-invariants. The condition that m shall touch a compoond curve rir will evidently be fulfilled if u tooch eitho- r <ir v, or go through an intersection of them. For an intersectioo counts, as has been said, as a double point on the oDiB|dez curve ; and a line going through a donUe point of a cnrre is to be considered doubly as a tangent. Henoe if T^ii, r] deoole the tact-invariant of k, r, we have when £ (u^ r, «r) is the resultant of k, r. ir. And the remit may be verified by comparing the order in whSdi tbe cdeffioests of lu r« or tr occur in these invariants. Thus, iar the cxxsSaata of K) we have TACT-INVARIANTS, 171 189. The theory of the tact-invariants of quaternary quantics is given in Oeometry of Three DimensionSj p. 544 ; and there is not the least difficulty in forming the general theory of the class of invariants we have been considering, to which Dr. Sylvester proposes to give the name of Osculants, Let there be t quanticS| Uj F, Wy &c. in k variables ; then the osculant is the condition that for the same system of values which satisfy Uj F, &c. the tangential quantics a??7/ + yt7j' + &c.,&c. shall be connected by an identical relation \(ajZ7/ + &c.) + /A(a?F/ + &c.) + v(a;TF/ + &c.) + &c. = 0. In other words, the osculant is the condition that the equa- tions Z7= 0, F= 0, &c., and also the system u„ c;, o;, &c. =0 r„ F„ F., &o. W,, TF., TF., &c. can be simultaneously satisfied. This latter system having k columns and i rows is equivalent to k- i+1 equations ; there- fore this system combined with the given i equations is appa- rently equivalent to A; + 1 equations in k variables. It is really, however, only equivalent to k equations ; for writing Z7= in the form a: fTj H- y D^ -f &c. = 0, and similarly for F, &c, we see that when the system of determinants is satisfied, and all but one of the equations Z7=0, F=0, TF=0, &c., the remaining one must be satisfied also. The system then being equivalent to k equations in k variables cannot be simultaneously satisfied unless a certain condition be fiilfilled. The order of this condition, in the coefficients of U^ is found by the same method as in Oeometry of Three Dimensions, We write for U^ U-\- Xw, and we examine how many values of the variables can simultaneously satisfy the i + 1 equations F, TF, &c., and the system equivalent to A; — 1 equations £7„ d;, d;, u I) u, a) U 89 F, F., F 1) S9 S) »F, TF, W. 19 29 89 &C. &C. &C. &C. 1 = 0. I 172 SYSTEMS OF QjJ ANTICS. The order of the i — 1 equations F, W^ &c, is the product of their degrees n, /?, &c. ; and the order of the osculant in thei coefficients of V is the product of this number by the order of the system of determinants, which is found by the rule given io a subsequent Lesson on the' order of systems of equations. When we are given but one quantic, the osculant Is the discriminant ; when we are given k quantics in k variables, the osculant is the resultant. The theorem of Art. 110 may be ex-i tended to osculants in general ; viz. that if we form the osculant of &— 1 quantics in k variables, and if the last be the product of two quantics Z7, F, then the osculant of tfee entire system will be the product of the osculant of the system of the ptl^er A; — 2 with U^ that of the system of A - 2 with F, and the square pf the resultant of all the quantics. 190. We have already seen (Art. 151) how the invariants Und covariants of a single quantic are derived from those of ^ system of quantics in the same number of variables ; and we wish now to point out how the invariaqts and covariants of ^ single quaqtic are connected with those of a system of quantics in a greater number of variables. Suppose, in fact, we had two ternary quantics, geometrically denoting two curves, we can, by eliminating one variable, obtain a binary quantic satisfied by the points of intersection of these curves ; and it is evident, geometrically, that the invariants of the binary quantic (exf pressing the condition, for instance, that two of these points should coincide, or should have to each other some permanent relatioq) must also be invariants of the system of two ternary quantics. Conversely, we may consider any binary quantic as derived from a system of two ternary qviantics; for we have only to assume X= <j> (a?, y), Y= yjr (a:, y), Z^= %(a;, y), equa^ tions which in themselves imply, by elimination of x and y, one fixed relation between X, F, Z^ and from which, combined with the given binary quantic equated to zero, we can obtain a second such relation. The simplest example of such a transfor- mation is that investigated by Mr. Burnside Quarterly Journal X. (1870) p. 211 ; (compare Conies^ note pp. 386-7), where ^, y^^ x BINARY CONNECTED WITH TERNARY SYSTEMS. 173 are q.uaclratic* functions of x and y. The substitution is then reducible by linear transformation to X=a?*, Y = 2a;y, -^=y', giving the fixed relation 4-XZ— F' = 0. By making these substitutions for x* &c. in a binary qnantic of even degree, we have at once a second relation between X, Yj Z; if the iquantic be of odd degree, it can be brought to an even degree by squaring. The resulting relation is obviously not unique, but is of the form 0^^ + 0j^., (4ZZ- Y^J, where <f>^^ is any one form of the relation, and the coefficients in (f>^^^ are arbitrary. Geometrically, the binary quantic of the w** degree is thus made to represent m points on a conic, determined when m is even by the intersection of the conic with a curve of the order ^?w, and when m is odd with a curve of order m touching the conic in m points. Among these forms there is always one whqse invariants and covariants are also invariants and co- variants of the given binary quantic.f Thus the binary quadratic ax* + 2bxt/ + cy' is replaced by the system aX4- JF+cZ, 4tXZ— F', and geometrically denotes the two points of intersection of a line with a conic. The dis- criminant of the quadratic is also an invariant of the system ; that whose vanishing expresses the condition that the line shall touch the conic. So, in like manner, the system of two binary quadratics aa?* + ibxy + cy*, aV + 2Vxy + c^ , gives rise to the system of a conic and two lines. The invariant of the binary system oc' + ca'— 266' (Art. 151) is also an invariant of the ternary system ; viz., its vanishing expresses that the lines are harmonic polars with respect to the conic. If three lines Z, M^ N be mutually harmonic polars with respect to a conic, we know [Conies^ Art. 271) that the equation of the conic may be written in the form F= lU + mM^ + nN^ = 0, whence we infer immediately that if three binary quadratics be * If linear functions had been taken, the transformation conld be reduced to X-=Xf Y=i/f Z=0, and the binary quantic of the n** degree would represent n points on the line Z (see Art. 177). t This form can be found by operating on <t>tm+<t>vi^-2 (4ZZ— F*) with the form reciprocal to 4irZ-r*, viz. y^y-^- jy»> ^^^ equating to zero the coefficients of every term in the result. 174 SYSTEMS OF QUANTICS. connected in pairs by the relation ac' -{-ca^ - 2lV = 0, their squares are connected by an identical relation t&*4-»wM'4- w-A7^==0, for V vanishes identically when we return to binary qnantics. To the Jacobian of two binary quadratics answers, for the system of two lines and a conic, the line 2(ai') X-\'[<ic') F+ 2(Jc') Z^ which is also a covariant of that system. In fact, it is the polar with respect to the conic of the intersection of the two lines. More generally, the Jacobian of any system u, v will be transformed into the Jacobian of the system formed by Z7, Vy and the fixed conic. For let u^^ u^^ u,, denote the dificrentials of u with respect to X, F, Z^ which, it will be remembered^ denote a:*, 2ajy, y^ respectively, then the Jacobian is y\-xy, :e = «» «»> ^ ".» ».» ". u \1 V 19 U, V. but the terms in the first line are proportional to the differentials of4X?-r*. The same method being applied to the discussion of the biquadratic, it is found to be equivalent to the system of two conies, viz. the fixed conic 4XZ'— Y^^ and the conic ax»+cr«+ez^ + 2irz+2czz+2jzr=0, the discriminant of the latter conic being also an invariant of the quartic (Art. 171). So again the system of two binary quartics is equivalent to a system of three conies. We shall have occasion in the next Lesson to give farther illustrations of this method : it has been applied by Mr. W. R. W. Boberts to the system of two cubics which involve properties of a twisted cubic, Proc, Lond, Math. Soc, vol. xiii. (1881), and the relation between binary and quaternary forms is developed with the general symbolic formulae in a paper by Dr. Linde- mann, Math. Ann. (1884) xxili. p. Ill, &c. / r ( 175 ) LESSON XVIL APPLICATIONS TO BINARY QUANTICS. 191. Having now explained the most essential parts of the general theory, we wish to illustrate its application by enumerating the different invariants and covariants of binary quantics for the lower degrees. K S and T be invariants of the same degree, or covariants of the same degree and order, and k any numerical factor, then S-hkT^ which will of course be also an invariant or covariant, will not be reckoned in our enumera- tion as distinct from the invariants 8 and T. And, generally, any invariant or covariant which can be expressed as a rational and integer function of other invariants and covariants of the same or lower degrees is said to be reducible, and will not be considered as distinct from these latter functions. It is otherwise if the expression be not rational and integer. Thus, if 8 be an invariant of the second and T of the third degree, then though 8^ + kT* would not be regarded as a new invariant, yet if it be a perfect square, and we have jB*= fi® + A;?^, we count £ as a new invariant distinct from 8 and T, and call it irreducible. It was proved in Art. 121 that a binary quantic has n — Z absolute invariants, and in Art. 1 22 that from any two ordinary invariants an absolute invariant can be deduced. We should infer, therefore, that the number of independent ordinary invariants is one more than the number of absolute invariants ; or, in other words, that a binary quantic of the n** order has w — 2 invariants, in terms of which every other invariant can be expressed. But as it does not follow that the expression is necessarily rational, we do not in this way obtain any limit to the number of irreducible invariants. And so as regards the covariants (including in this expression the invariants) we shall presently see that for a quantic of the w** order there are, inclusive of the quantic itself, n covariants, such that 176 APPLICATIONS TO BINARY QUANTICS. every other covariant multiplied by a power of the qaantic is equal to a rational and integer ftinction of the n covariants ; thus, each such other covariant is a rational, but not an integer function of the n covariants; and we do not hereby obtain any limit to the number of the irreducible covariants. We have stated (Art. 145) the method by which Prof. Cayley originally attempted to determine the number of distinct covariants and invariants. He did not at the time succeed in obtaining any limit to their number for quantics above the fourth order. Sub- sequently Gordan proved (see Oe?fe, vol. LXix., or Clebsch, Theorie der bindren algebraischen Formerly p. 255, also his Pro- gramm for the University of Erlangen, Ueher das Formensystem bindrer Formerly Leipzic, 1875), that for a binary qaantic or system of binary quantics, the number of distinct invariants and covariants is always finite ; and he has given a process by which when we have the complete system of invariants and covariants for a qnantic of any degree, we can find the system corresponding to the next higher degree. His proof, which is founded on an analysis of the different possible expressions by the symbolical method explained Lesson XIT., will be found in a subsequent Lesson on that method. Later still. Prof. Sylvester has investigated the whole subject by Prof. Cayley's method, founded on the theory of the partition of numbers, in various memoirs in the Comptes Rendusj vol. LXXXiv. pp. 974-5, 1113-6, &c., and subsequent volumes, as also in the American Journal of Mathematics. 192. It will be convenient to bear in memory what was proved, Art. 147, that a covariant is completely known when its leading coefficient, or, as we have there called it, its source, is known; this coefficient being any function of the differences of the roots of the quantic* Thus take the quantic (a, J, c...3[a;, y)", we know that, in the case of the quadratic, {ac — h^) is an invariant ; and if we desire to form the covariant ^^^■^ ■■ ■ ■■■■■■ ■■■■■■ ■»■■■ ^■■■■■■■» ■■— ^B^— . I I \m^^m^,^m^^m^^^mm * Such functions have been called semi-invariants or seminvariants, as they remain unaltered (see Art. 62) when we substitute x + \ for a;, but not necessarily when we substitute ^ + X for y ; and as they satisfy one of the difEerential equations given in that article, but not necessarily the other. INTEGRALS OP PARTIAL DIFFERENTIAL EQUATION. 177 {ac -^V) a? + &c., having this leading term, we observe that' the weight of the given source and its degree in the coeffi-? cients are each =2, so that writing 6^2 in the formnla (Art. 147) 2 = i(n^-^) we have j? = 2(w-2). The other coefficients are found by the method explained in that article } thus the covariant is found to be [ac - V) a? ^"-»' + {n - 2) [ad - be) x'^'^y It follows also from what has been stated in the article referred to, that any algebraic relation between the sources of different covariants implies a corresponding relation between the cor variants themselves. Prof. Cayley has used this principle in attempting to forni the complete system of the covariants of a binary quantic; and though it does not lead to any general theory it furnishes the most elementary and satisfactory proof of the numbers of concomitants for functions of the first four degrees. The leading coefficient of any covariant being a function of the differences must (Art. 62) satisfy the differential equation (arf^ H- 2M. + 3ccZ^ + &c) Z7=0: and we assume that 27 is a rational and integer function of a, 6, c, &c. Now, if we solve the partial differential equation, we find that U must be a function of a, ac - 6*, a^d - 3a6c + 2&', a^e - \ofld + 6a JV - 3 J*, &c. , where the law of formation of the successive terms is obvious ; and, in fact, the covariants of which these terms are the leaders are each the Jacobian of the preceding covariant in the series, combined with the original quantic. We shall refer to these quantities as X,, X,, Z,, &c. and we see that the leading coefficient of any covariant must be a function of these quanr titles: and it must of course be a rational function of them* The question is whether there are any rational, but not in- teger functions of Z^, Z,, L^ &c., which are rational and integer functions of a^h^c\ and a little consideration shows that the only admissible form is that of a rational and integer function divided by a power of Z^, that is a. For, the leading AA 178 APPLICATIONB TO BINARY QUANTICft. coefficient in question is a rational function of the coefficients (a, i, c, ...] ; and if we make in it i = 0, it becomes a rational function of a, c, d^ &c.| and by multiplying by a suitable power of a it can be made an integer function of a, aCj a^dj a\ &e« But these are the values of X,, X,, &c. on the supposition of 6 = 0. Thus we see that the leader of any covariant can only be the quotient by a power of a of an integer function of these ti quantities. Conversely, the problem of finding all possible covariants is the same as that of finding the new functions which arise when rational and integer functions of Z,, X,, &c. are formed which are divisible by a. To find these functions We make a = in X,, X,, &c. and eliminate b between any pair ; we thus get a function of X,, &c. which vanishes on the supposition of a^Oj and therefore is divisible by a power of a. By performing the division we obtain the leader of a new covariant. This again may be treated in like manner, by putting a = and examining whether it be possible to eliminate the remaining coefficients. This method will be better understood from the applications which will be made presently. It is obvious that the same considerations apply to the still simpler forms — of lowest degrees — of particular integrals of the partial differential equation a, ac— 6',o*rf— 3aic+2i', ae—4tbd+ Zc\ ay— 5aie + 2ac<?— 66c' + 8J*rf, &c. of which the second, fourth, &c. are the successive quadrinvariants of even quantics as they arise, and the third, fifth, &c. are the sources of the evectants of the successive quartinvariants of the corresponding odd quantics as they arise. See Art. 142. 193. We have already stated the principal points in the theory of the quadric form (a, J, c}[a;,y)*. Since there are but two roots and only one difference, there can be no function of the differences of the roots but a power of this difference; and the odd powers, not being symmetrical functions of the roots of the given quadratic, cannot be expressed rationally in terms of its coefficients. It thus immediately follows that the quadric has no covariants other than the quantic itself, and no invariant other than the powers of the discriminant, ac-^b^ SYSTEM OF TWO QUADRICS. 179 w^ieh is proportional to (a — )3)*. We have already shewed (Art. 157) that it follows, by Uermite's law of reciprocity, that only quantics of even degree can have invariants of the second order in the coefficients. These are the system whose symbolical form is 12% explained Art. 153, ac - J>*, at - ihd + 3c', ag - 6 J/+ 15ce - lO^f, &o. If we make y = 1 in the quadratic it denotes geometrioally a ayitem of two points on the axis of x, and the vanishing of the 4iscriminant expresses the condition that these points should filicide. Art. 177. System of two quadrics. This system (a, J, cXxj y)\ [a\ b\ cjxj y)% ^as the invariant 12' or ad + ca''' 2bb\ When each qnantic is taken to represent a pair of points in the manner just stated, the vanishing of this invariant expresses the condition (see Conies^ Art. 332) that the four points shall form a harmonic system, the two points represented by each quantic being con- jugate to each other. We have also proved (Art. 177) that the covariant 12 (or the Jacobian of the system) represents the foci of the system in involution determined by the four points. It is easy to see, as in Art. 169, or by Conies^ Art. 333, th^t the Jacobian may be written in the form J[u^ v) = y%-xy^x^ a a' b , c Now by the ordinary rule for multiplication of determinants we have y, - xy, 'J? a, b ^ e X a; b', o' 0, u V Uj 2i>, A V, A , 2D' c , — 2 J , a o\ -2b\d or 2«7' = - 2M'i)' + 2mv A - 2v'2>, where J denotes the Jacobian, D and D' the discriminants of the quantics, and A the intermediate invariant ac' -i- ca' — 2&i'« liW APPLICATIONS TO BINARY QUANTICS. This equation includes the theorem stated Art. 190, for the case A = 0. The equation just given may also be easily verified by means of the canonical form. We have seen (Art. 177) that there are two values of k^ for which U'{-iv is a perfect sqnare, and if these squares be x^ and y* the system may be written aa;' + cy*, a^x^-hc'y'j or more simply a5*+y', ax^-\-cy^. We have then -D = 1, 1/^ac^ ^^a-\-c^ /= (c — a) xt/j by means of which values the preceding equation is at once verified. So again the Jacobian of u, J is for the canonical form ]^ (c- a) (a^ — y*), and therefore is in general ^Au — Dv. The invariant A taken between u and J vanishes identically, as is geometrically evident. All other invariants or covariants of a system of two qnad- rics may be expressed in terms of w, v, J, Z>, iX, A, Thus the eliminant {ap' - caj + 4 (5a' - aV) {he' - cb'), may also be written in the form {ouf + ca'- 2hby - 4 (ac - b*) (aV - J"). In other wordg, the eliminant is the discriminant either of the Jacobian (aJOa^+K)iry+(JcV, or of {ac - V) V + {ac' + ca' - 2bV) \fi + (aV - ^ fi\ The former expression is linearly transformed into the latter by the substitution \b + /aJ', - (Xa + /*«') for x and t/. System pf three quadrics {a,h,cjx,y)\ i:a\V,cJx,y)\ {a:\V',c'Jx,y)\ This system has, in addition to the invariants and covariants corresponding to the respective pairs of quadrips, the dete^r minant 23.31.12, a, hj c R = a', S , c' a ^ ^ c whose vanishing expresses the condition that the three pairs of points represented by the quadratics shall form a system in involution (Ex. 7, p. 25, a]so Conies^ p. 310). This invariant THREE OK MORE QUADRICS. el81 formed for u^ v, J is another expression for tbe eliminant of u an4 V. The expression found for J* of two quadrics may be generalized, if in tbe second determinant we write in the second and third rows c", -2^', a"; c"', -26% a'". We thus find that if there be four binary quadrics m„ w^, w,, m^, 0, w, , w. 2e/;A= We get, similarly, an expression for the product of two invariants B^^R^^ 2R R = ^4? - 2 J„ a, ^6? - 2*51 «5 ^,41 A», Ae ^.1 ^., ^. ^84, ^85, ^ae To these formulae maj be added tbe following, tbe tratb of wbicb is easily seen, M, J„ + «/„ + «/„ = 0, M, J?^ - "A. + «A„ - W4^,„ = 0, From the linear relations connecting the t^'s and e/'s follow the quadric identities = u i) U 89 U, S9 Al) ^,8) A») ^1 Ao ^«J ^«J «*. . AlJ ^88) ^88J ^Z Thus a system of three binary quadrics at once gives rise to a conic and their three Jacobians to its reciprocal fomi. The equation of the conic is referred to any line and two others through its pole when the three binary quadrics are any two arbitrarily taken and their Jacobian. The . conio breaks up into right lines if the three binary quadrics form nn iavolution. 182 APPLICATIONS TO BINARY QUANTICS, 194. It is to be remarked that, by means of Euler^s theorem for homogeneous functions, the theory of those covariants of any quantic, the expression of which contains differential coefficients in not higher than the second degree, reduces itself to the theory of the quadric; and so every relation between the covariants of a quadric has answering to it a relation between such covariants of a qnantic in general. Similarly, a relation between covariants of a cubic gives a relation between general covariants not involving differential coefficients in more than the third degree, and so on. Thus, the expression obtained for the square of the Jacobian of two quadrics gives the identical relation {{ax + by) {Vx + c'y) - (a'« + Vy) [hx + cy)Y ^^{ac" V) (aV + 2Vxy + cV)' + &c. But if a, 5, c ; a\ h\ d denote the second differential coefficients of any two quantics, we have whence we have an expression for the square of the Jacobian of any two quantics + nn' (« - 1) (n' - 1) Awt; -rf{7i- 1)* JI V, where E. denotes the Hessian ac — V and A, as before, the CO variant ao^ -hca^ — 2bb\ So again, since the Jacobian involves only differential co« efficients in the first degree, the Jacobian of cT, u^ involves them only in the second, and therefore can be expressed by means of the theory of the quadric. Writing i, M for the first differential coefficients, we have J[J, u) = {aMif'- J(iM'+i'Jf j + ciiVK^""^ -2J'Zif+c'i*}. But the values of the two members of the right-hand side of the equation are immediately found by the canonical form of the quadric, and are respectively ^^' ZT J ^ A w'(n'-l) „ tHv. and 7AU — -^ — -~ Hv. »— 1 ' n — 1 l^""!} THE CUBIC. 183 whence J (J* u) = — 775 — - Hv Aw. ^ ^ ' \n — iy n - 1 195. The cubic. We come next to the concomitants of the cubic Z7= (a, J, c, djx^ y)\ It has but one invariant (Art. 167), viz. the discriminant i> = a*rf* + 4ac' - 6a5ci + 4d&» - 3iV. If the cubic were written without binomial coefficients, the discriminant would be 27ay* + 4ac*- 18aJci+ 4di'- JV. It is to be noted that the function here written is, with sign changed, the product by a^ of the squares of the differences of the roots of the cubic. A useful expression may be derived from the last remark. Consider the three quantities iS — 7, 7 — a, a — )3, they are the roots of a cubic for which a = 1, 6 = 0, c— 4{(/3-7r+(7-a)"+(a-/8rj, rf=03-7)(7-a)(a-/8). Hence {2a - /8 - 7)' (2^ - 7 - a)» (27 - a - j3)« = 4{(i8-7r+(7-a)"-Ka-/8)y-27(/3-7)'(7-ar(a-/8)\ The Hessian 12* or H^ is hj Cj d y\ -iry, »" This has the same discriminant as the cubic itself (Art. 167). The cubic covariant 12M3, or the evectant of the discrimi- nant, which we call e7, is (see Art. 142) (aV-3aJc+26', a&i+i'c-2ac'', 2JW-ac^-Jc', 3 J<xf-ad'-2c'3[a?, y)', which may also be written in the determinant form x\ Zofy, Zxy\ y» c, - 2J, a, rf, — c, " b^ a 0, dj -2c, b This cubic may be geometrically represented as follows : — If we take the three points represented by the cubic itself, and take the fourth harmonic of each with respect to the other two^ we 184 APPLICATIONS TO BINARY QUANTICS. get three new points which will be the geometrical representa-^ tion of the covariant in question. This theorem is suggested by its being evident on inspection, that if the given cubic take the form afy (a? + y), then x-^y will be a factor in the covariant, as appears by making a = d—0^ J = c = l in its equation. But a? + ^j a? — y are harmonic conjugates with respect to x and y. Now, if a, )9, 7, 8 denote the distances from the origin of four points on the axis of a?, any harmonic or anhartnonic relation^ between them is expressed by the ratio of the products (a-/3) (7-S) and (a -7) (iS-5): and this ratio (see Art 136) Is unaltered by a linear transformation ; that is, when for each distance a we substitute r^ ,, Such relations, then, being unaltered by linear transformation, if proved to exist in one case, exist in general. We find that the other factors in the evectant of ary (a? + y) are x-k- 2y, 2aj + y, so that our result may be written symmetrically, that the evectant of xyz (where 0?, y, z are connected by the linear relation aj + y+a = 0) is (y — «)(« — «) (a;-y). These considerations lead us to the ex- pression for the factors of the covariant in terms of the roots of the given cubic : for if h be the distance from the origin of the point conjugate to a with respect to /8 and 7; solving for S from ^, .2 1 1 ^5, a/S -4- a7 - 2^87 the equation __g = _^ + __ we get S = -^_^__, whence the covariant must be a' {(2a- /3 -7) a; + (2/37- a)8-a7)y} {(2^8- a-7) 0? + (27a-/S7-i8a)y}{(27-a-/8)a;+(2a/3-7a-7^)y}=-27/, as may be verified by actual multiplication and substitution In terms of the coefficients of the equation. * The anharmonic ratio of four qnantitles his any of the aix Talnes, aoooidmg to the order assumed, X, r- , 1 — X, ^,-^y i — r — » v _ , » which are in geneial all different. They may .come to have equal values either if X = 1 when ttio Talues of the quantities are equal and the other values of the anhaniKxiic ratio are and oo; or if X = — 1, whoi the quantities form a harmome sene^ and the other values of the anharmonic ratio are 2 and |; or if X* — X + 1=:0, when the quantities form an equi-anhaniMnic series, three values of the anharmonic ratio are one imaginary cube root of — 1, and three its other imaginary cnbe root. THE CUBld. 185 Similarlj for the quadric covariant, if o) be an imaginary cube root of unity and we solve for 8 from we get (a + coiS + ©'7) S + /37 + W7a + fi)'ai8 == to determine a distance equi-anharmonic to a, /S, 7. Hence^ W6i have the covariant {(a + (afi + a)*7) x + (^87 + taya + co'a/S) y] X {(a + to"yS + 6)7) X + (/S^7 + o)'7* + ^^ifi) y}, dotlble this is found to be = S (a; — a)' (y8 — 7)', which, multi-* plied by a' and expressed in terms of the coefficients, = - 18^7. Id6. We can now see that our list of covariants is complete^ The leading coefficient of any covariant is a function of the differences /8 — 7, 7 — a, a — ^. Since the sum of thesef quantities is zero, any symmetric function of them can be expressed in terms of the sum of their squares, and their continued product. But since this product is only half sym*^ metrical with respect to the roots of the given cubic, that is to say, is liable to change sign by an interchange of the roots, it can enter only by its square into a function expressible in terms of the coefficients. We thus see, that if the leader be a symmetric function of the differences, the covariant can be expressed as a rational function of Uj E^ D. But there is another function, viz. the product of the dif-^ ferences (2a — /8 — 7) (2^8 — 7- a) (27 — a — /8), which though only half symmetrical with respect to the differences, is symme- trical with respect to the roots of the given quantic. This is the leading term of the covariant J. But obviously the squarer of this function can be expressed in terms of the sum of squares^ and product of differences. The expression has, in fact, been given in the last article. It is easy to prove, that in the case of the cubic written with binomial coefficients, we have a«S(a-y3)« = 18(y-ac), a^(^-7)'(7- a)'(a-/3)'^-2?i>, a' (2a- /S- 7) (2/3- 7-a) (27-a- /3) =:- 27 (aV- 3a6c+ 2i')^ BB 186 APPLICATIONS TO BINARY QUANTICS. bj the help of which values, the expression obtained in the last article gives the relation between the covariants, due to Pro£ Cay ley, This relation may also be easily verified by using the canonical form U= ax* + dy*^ in which case we have D = a*(Z", -5"= adxy^ J=^ ad {aa? — dy*). Any other relation between co variants may be similarly investigated. Thus we can prove that the dis« criminant of / is the cube of the discriminant of U^ the former discriminant being for the canonical form ct^d^. So again we see that the Hessian of / differs only by the factor D from the Hessian of U. Prof. Cayley has used the relation just found between eT", 2), J7J and -ff, to solve the cubic J7, or, in other words, to resolve it into its linear factors. For, since J* — Z>Z7* is a perfect cube, we are led to infer that the factors J±UsfD will also be perfect cubes, and, in fact, the canonical form shows that they are 2a*dx* and 2ad'^y*, Now, since xa^-\^ydh is one of the factors of the canonical form, it immediately follows that the factor in general is proportional to a linear function which evidently vanishes on the supposition 1^=0. Ex. Let us take the same example as in Art. 166, U= 4rr* + 9x^ + ISxy* + 17y*. Here we have J) = 1600, /= 110a:» - 90x^y - 6S0xy^ - 670y«, whence U43 + J=10{3x + yy; U^- JzzbO (x + By)^-, and the factors are Bx + t/+{x + By) »J6. 197. The entire system of covariants for a cubic is also im- mediately found by Prof. Cayley's method explained Art. 192. We start with the three covariants Z7, Hj J^ whose leading coeffi- cients are L^ — a^ L^ = ac—b'*j Lj^ = a^d—3abc + 2b\ If we make a = 0, the last two become — i*, 26', whence by eliminating b we have 4jt/+ -L * = 0. Thus we see that 4:H*-\- J^ is divisible by a, and actually it is found to be divisible by a', the quotient being D or a'-^cZ''' + 4ac' + 4rf6' - 3&V - GaJcrf. We have thus obtained the new invariant D, together with the equation of SYSTEM OF CUBIC AND QUADRIC. 187 connection 4fl '+ /' = 2) V\ If in D we make a =* It becomes Adb^ — SVc*^ and since this combined with the preceding gives rise to no new relation between L^^ L^^ jD, we learn that the system of coVariants is complete. 198. System of cubic and quadric. Let these be U= [a, b, c, dXx, yf', F= {A, B, C^x, y)' ; then the following is a list of the different independent covariants of the system. The figures added to each denote its order in the coefficients of the cubic and in those of the quadric. Three cubic covariants^ viz. the original cubic Z7, (1, 0); its cubicovariant (3, 0) which we call e7, printed in full Arts. 195, 142, and the Jacobian of ?7, F, (1, 1) which is {Ab-Ba) x%{2Ac-Bb- Ca) x^y-\-[Ad'^Bc'-2 Cb) xf^Bd - Cc) y\ Three quadric covariants^ viz. the original quadric F, (0, 1) ; the Hessian of the cubic (2, 0} and the Jacobian of these two (2, 1) which is [A {ad^bc)--2B{ac--V), A {bd- c')--C{aC'- J'), 2B [bd ^c')--C {ad ^ bc)Jx, y)\ Four linear covariants^ viz. i, (1, 1) which is obtained by substituting differential symbols in the quadric and operating on the cubic, L^^[aC-2bB-^cA)x-\-[bC-2cB+dA)y\ 7/j (1, 2) which is obtained by operating in like manner with //, on the quadric, i, = [aBC- b (25* + ^ 0) + ZcAB - dA'} x + {a(7" - 3JJ5(7+ c [A (7+ 2J5») - dAB] y, and X3 (3, 1), and L^ (3, 2) which are obtained in like manner from the quadric and the cubicovariant c7, and which may be written at length by substituting for a, J, &c. in the values of i/,, ig just given, the corresponding coefficients of e7. Five invariants^ viz. A (0, 2) the discriminant of the quadric, i>(4, 0) that of the cubic, 7(2, 1) which is the intermediate invariant between the system of two quadrics F, H I^A{bd- c') ^Biad- be) ^G[ac^V), 188 appligahons to binabt quahtigs. B (2, 3), the resultant of the cnbic and qnadric, which formed by the methods of either Art. 67 or Art. 86, is ^9b'AC*'-lSbcABG+QbdA{2B^-^4C) and, finally, -M"(4, 3), the resultant 0ither of Z,, If^y or of i„ i, is Jf = a'dC- Sa^bcC'^ Qa*bdBG^+ QaVBG^+ 2ab'G'-{- Bab'oBO^ + Sab'dA G'+ UaVdB'G^&abc'A G'-24,abc'B'G-\-12ac'ABO + 8ac'5'-3ac'rf^»(7- 12acW^5"+ Qacd'A'B-ad'A^^^b^BG^ 4 Bb'cAG^ + Ub'cB^G- Ub^'dABG-Hb'dB''-^ ^VcdA^O + 24ycrf^J?' - 6ra'^*5-. 3Jc»^»(7-. Ubd'AB^ - 6JcW4«JS + 3 Jc^*^' + Qc'A^B - 2cV^'. This last Invariant ilf Is a skew invariant (see Note, p. 131) and changes sign if we interchange x and y ; the functions J^ 2/,, L are also skew functions. In comparing different invariants wq may conveniently make A and (7=0, which is equivalent to taking for x and y the two factors of the qi;adric. In this case the fundamental invariants are A = - -B», jD = a^d^ + 4ac' + 4cZ5' - 3 JV - 6a5c J, Z=-5(a(?-Jc), .Ii=:^-8B'ad, M=^BB\a(?'-caf^), Thus we have in the same case L,=^^2B{bx^cy), L,^^%B^{bx^cy), and L the resultant of these two is — SjB'^Jc, whence we see immediately that L can be expressed in terms of the funda? piental invariapts; in fact, L=^B + S/:^L So, ^gain, we see that the square of M can be expressed in terms of the oth^ invariants, giving a relation between them. For we haye 8 (ac' + db') = ? (i) - a»(Z* + 3 JV + Qabcd)^ whence if '^ = 4JS« (jD - a'd"" + 3iV + 6aJcrf)» - 256-B«ae»V, and if in this equation we substitute for cm?, ^-^ , for 5c, — p, , THE QUABTIC. 189 and for 5', - A, we have the reqaired relation J/" = - 4tA'D' + i) (J2* 4 12R^I+ 24.AU') - 4BJ' - 36 A/\ Geometrically, see Art. 190, if the cubic U be represented by three points on a conic its J covariant determines on the same conic the harmonic conjugates of each of the three with respect to the other two; the H covariant determines the double points of the involution of these six points. Or we may state it thus (see Conies^ p. 387), the triangle touching the conic at the vertices of U is in perspective with [7, the lines connecting corresponding vertices mark oflF J on the conic and intersect in the centre of perspective 5„ the axis of the perspective meets the conic in the points H : H^ is also the pole of ZT with respect to the conic. Any quadratic F gives a right line meeting the conic in two points, and the line joining its pole to the centre of perspective ZT, is the Jacobian of H and V and is the axis of a new perspective whose centre is on the conic and given by the linear covariant -C,. The line joining i, and the pole of H meets the conic again in ig. ij is the harmonic of i, with respect to F, and L^ is the point where H^L^ meets the conic again. The invariant / vanishes for any right line which passes through H^. 199. The quartic. We come next to the quartic, which, as we have seen, pp. 128-9, has the two invariants S=ae- ibd + 3c' and r= ace + 2hcd - ad^ - e5' - c\ We have shown (Art. 172) that the quartic may be reduced to the canonical form x* + 6mx^i/* + ^*, and for this form these in- variants are ;S'= 14 3m', T=m — vi\ These invariants, expressed as symmetric functions gI the roots, are 24/S= a'2 (a - /3)' (7 - S)% or 125 = a« {jSy + a3 + o» (ya + fid) + cu^ (a^ + yd)} {fiy + aS+ (o« (ya + /33) + cu (afi + yd)}, also — = {(«-P)(y-i)-(«-y)(i-,9)}» + {(«-y)(i-/3) -(.-«) (^-y)}« + l(a-e) (/3-y)-(a-)8)(y- «)}«, and 4322'=a'2(c«--/3)*(7-S)'(a-7)(/3-8), or, more can-- Tenientlji 190 APPLICATIONS TO BINARY QUANTICS- In the latter form it is easy to see that T^ is the condition that the four points represented by the quartic should form a harmonic system, thus T may be called the harmonic invariant of the quartic, and in like manner S its equi-anharmonic in- variant, see Note, p. 184. It was stated (Art. 171) that r=0 is the condition that the quartic can be reduced to the form Q^-\y^^ and that Tcan be expressed as a determinant S, c, d If A be the modulus of transformation, then (Art. 122) B and 3r become by transformation A* 5, A'jT, respectively; and the ratio 8^ : T* is absolutely unaltered by transformation. ^ 200. To express the discriminant in terms of 8 and T, It has been already remarked (Art. Ill) that the discriminant of a quantic must vanish, if the first two coefficients a and b vanish ; for, in that case, the quantic, being divisible by ^', has a square factor. On the other hand it is also true, that any invariant which vanishes when a and h are made = 0, must contain the discriminant as a factor. Such an invariant, in fact, would vanish whenever the quantic had any square factor {x — ay)* ; for, by linear transformation, the quantic could be brought to a form in which this factor was taken for y, and in which therefore the coefficients a and J = 0. But an invariant which vanishes whenever any two roots of the quantic are equal, must, when expressed in terms of the roots, contain as a factor the difference between every two roots ; that is to say, must contain the discriminant as a factor. It is easy now, by means of 8 and T, to construct an in- variant which shall vanish when we make a and J = 0. For on this supposition 8 becomes 3c^, and T becomes — c® ; therefore i8®--272" vanishes. Now this invariant of the sixth order in the coefficients is of the same order as that which we know ♦ Dr. Sylvester gives the name catalecticant to the invariant, which expresseB that A quantic^of order 2n can be reduced to the sum of n powers of the degree 2n. DISCRIMINANT OF A QUARTIC. 191 (Art. 105) the discriminant to be. It most therefore be the discriminant Itself, and not the product of the discriminant by any other invariant. The discriminant is therefore {ae - 4J>d + 3c')' - 27 {ace + 2bcd - oci" - eJ* - c')*. We can in varions ways verify this result. For instance, it appears from Art. 185,^ that the discriminant of the canonical form X* + 6mx*y* + y* is the square of the discriminant of the quadratic x* + Qinxy + y* ; that is to say, is (1 — dm^)\ But (1 - dmj = (1 + SmJ - 27 (m - m')\ We should also be led to the same form for the discriminant| by writing the quartic under a form more general than the canonical form, viz. Ax*-^ By^ + Cz*^ where x + y + z = 0. In this case we have a = A+Cj e = B-\-C^ h=:c = d=Gj and we easily calculate 8=^BC+CA + AB^ T^ABO. But if we equate to nothing the two differentials, viz. -405* - Gz\ By^ — Cz^ we get a?', y®, z^ respectively proportional to BCj CAj AB; and, substituting in x + y + z = Oj we get the dis- criminant in the form {BC)i + {CA)h + {AB)h=.0, which is {BG-\'CA'{-ABy^27A'B'C'=:0 or /S'-27T' = 0. 201. From the expression just given for the discriminant of a quartic in terms of S and T can be derived the relation (Art. 196) which connects the covariants of a cubic. If we multiply two quantics together, the invariants of the compound quantic will be invariants of the system formed by the two components. If then we multiply a quantic by x^ 4 yrj^ the invariants of the compound will (Art. 134) be contravarianta of the original quantic ; and when we change | and rj into y and — 0?, will be covariants of it. If we apply this process to a cubic, the coefficients of the quartic so formed will be <^I/i i[^hy-ax) iicy-bx)^ {{dy-Scx)^ -dx; * We may also see this directly, thus : The resultant of oic* + 6y*, o'a* + i'y* is the k^^ power of ab' — 6a', since the substitution of each root of the first equation in the second gives ab' — ba\ [Now the discriminant of oar* + Qcx^y^ + cy* is the xesultant of asc^ + 8cxi/^f Sex^y + ey^. If we substitute a; = in the second, and y = in the first, we get results e, a, respectively, and the resultant of ax^ + Bcy^, 3«r* + ey* is {ae - 9c2)», xhe discriminant is therefore ae {ae - 9c«)». 192 APPLICATIONS TO BINARY QUANTICS. and the invariants 8 and T of this quartic are found to be the covariants — Jfl, j\J of the cubic. But the discriminant of the product of any quantic by x^ + yrj^ by Art. 110, becomes, when treated thus, the discriminant of £7, multiplied by U\ Express-* ing then the discriminant of the compound quartic in terms of its 8 and jT, we get the relation connecting the ffj J^ and discriminant of the cubic. 202. A quartic has two covariants, viz. the Hessian My whose leading coeflScient is ac — J*, and / the Jacobian of the quartic and its Hessian, whose leader is aV— Sabc + 26*. The Hessian is the evectant of jT, its value is H=^ {ac - J') x* + i{ad- be) ^y + (ae + ibd - 3c*) ccy + 2 (ie - erf) i/-f (ce-rf*)/ or 3fl = a, J. I rf c rf c rf rf 3/, -2a'y, a' e 0, y 35, -a rf, —3(7, e , — 3rf, , a?" , 2a;y, ^ 3c, y\ , — 2a?y, 3x and becomes, for the canonical form, 7n(a;* + 2/*) + (l-3m*)a:y. Expressed in terms of the roots, it is - 48fl^=.a'S (a - ^f [x - 7)* {x - l)\ The covariant eT", which symbolically is 12" 13, written at length, is J^ [a^d" Sabc + 2b\ c^e + 2a5rf- 9ac'*+ 6J'c, 5aie- 15acrf+ lOiV, - lOarf" + 10i% - 5ac7c + 15Jce - lOW, - ae' - ibde + 9c'e - 6crf", - be^ - 2rf' + Zcde^x^ y)* = i a* , 2^x\ 2^xy% y" , , aj' , 3a;''y, 3a:3^», / — rf, 3c , — 3J, — c 2rf, , -2J, a , - c , 3rf , - 3c, 6 and for the canonical form (1 — 9m'} xy [x^ -^*)» fcOVARIANTS OP A QUABTIC. 193 We have just seen that the x and y of the canonical form which we use are factors of /, but it will be remembered (Art. 172) that the problem of reducing a quartic to its canonical form depends on the solution of a ctibic equation ; hence, the factors of J are the x and y of the three canonical forms* This may be connected with the theory explained in Art. 177. If U and V are any two quartics, six values of \ can be found, such that 27+ XF shall have a square factor, and those six factors are the factors of the Jacobiatn of U and F. But when V is the Hessian of 27, the sextic in question becomes a perfect square, and there are three valties of \, for each of which U-\-\V contains tioo square factors, but these factors are still the factors of the Jacobian of U and F. The geometrical meaning of J may be stated as follows : let the quartic represent four points on a line Ay J?, C, Dj then these determine three different systems in involution (according as B^ G ov D is taken as the conjugate of A)^ and the foci of these three systems are given by the covariant J, From the last remark we can express the factors of J in terms of the roots. In fact, by Ex. 7, p. 25, the double points of the involntion formed by yS, 7 ; a^ h are determined by the quadratic = 0. Similar equations determine a; , 2a;, 1 ^7^ /3 + 7, 1 a8, a + S, 1 the foci for the other two systems. Ex. 1. To break up the quartic into two quadratic factom. Let {px^ + 2qxy + ry') (/>'x* + 2q'xy + r'y^) be identified with the quartic, and substitute in their places for jpp' = 0, pgt + qp* = 2d, gi*' + rg' = 2c?, pr* + fy' = 2 (c + 2^), ^g' = c — p in the identity tt' — e. as in Art. 25. The redtidng cubie is found to be Pf P r. r' a, Pf P g'f 2 r'f r ft, e-pj = 0, expanded tf + 2p d c + 2/t), dy Now, when we write (y - a) (^ - a) - (a - ffi (y - ^) = 12pi, (a -/3) (y - ^) - 03- y) (a - a) = 12p„ (/3 - y) (a - ^) - (y - a) (P - d) = 12^, in the expressions for 8 and T in terms of the roots, Art. 199, they become ^ = - 4a2 (/)2P, + pa/o, + pijog), r = - 40»/I)l/>2P8' Hence, since P\ + Pt + pi=^^i ^Pu ^Ptt ^Pz m® the roots of the cubic cc = or 194 APPLICATIONS TO BINAEY QUANTICS. Ex. 2. To discnaB the relations between the qnadiatic factors of /. Writing out the determinant forms, let ns call u = {fi+y-a-d)x'-2{fiy-'a6)xy-k-[fiy{a + d)'-ad{fi+y)]y^ = a^ae^2bixy+ciy*^ v=(a + /3-y-a)a»-2(a/3-yd)zy+[a/3(y + e)-yi(a + /3)]y«=a,a«+26,afy + Cay^. We have thos r-«^=-2(/3-y)(a^-ay)(a^-^y), w-«=-2(y-o)(a^-/3y)(aJ-^y), «-«=-2(a-/3)(a?-yy)(a?-^y), v+w=i 2(a-^)(aH3y)(aJ-yy), v>+u= 2(fi-6){x-yy){x-ay), t^*-v= 2{y-S){x-ay){x-fiy). Hence = = = — . Pt-Ps Pz-P\ Pi-Pt ^ Thus it appears at once that the identical relation (compare p. 181) between «, V, w is /ojtt* + /Ojr* + ^,w* = 0. Hence, as this relation involves only the squares, the qoadratics are harmonic in pairs. The same thing is found by actual calculation : OiCj + CiOj — 2ftid, = 0, &c., also a^c^ - V = (y- «) 03- a) (a-/3) (y -6) = 16 0>,-/t>,) (pi-p^, Ac, or, writing A = (/o, - p,) (/Oa - Pi) (/»i - Pa), («i<?i - *i*) C/»2 - Ps) = Ac. = 16 A. The value already given f (»r 1/ in terms of the roots may be written a« (f«* + v^ + w^=- 48£r. Combining this with the values of U given above we get ,^ ^ a2t»2 + 16£r o«i;« + 16JJ a*w* + 16JJ Pi P2 Pz £!x. 3. We have seen that J can differ only by a numerical factor from the t)roduct of the three quadratics ti, v, to. To determine it We may compare the leading terms of the two forms, or, expressing the symmetric function in terms of the coefficients, find that o»(^+y-o-^)(y + a-/3-^)(a + /3-y-a) = 32 (a«rf - Babe + 2*»), Whence a^uvw = 32 J, 203. Solution of the quartic. This is the same problem as that of the reduction of the quartic to its canonical form ax* + 6ca;y 4 ey^j for in this form it can be solved like a quadratic. One method of reduction has been explained (Art. 172); the reduction may also be eflfected by means of the values given for S and T. Imagine the variables trans- formed by a linear transformation whose modulus is unity, and so that the new b and d shall vanish; then we have 8=ae-{-3c*j T=ace-'(?\ and the new c is given by the equa- tion 4c^- Sc-\- r=0. We get the x and y which occur in the canonical form from the equations Z7= ax* -t Qcx^y* + ey*^ H^ acx* + {ae — 3c'} x*y^ + cey*, whence c U— H=^ {dc^ - ae) x^y\ SOLUTION OF THE QUARTIC. 195 Our process then is to solve for o from the cubic just given, and with one of the values of c to form cU—H which will be found to be a perfect square. Taking the square root and breaking it up into its factors we find the new x and t/^ and consequently know the transformation, by means of which the given quartic can be brought to the canonical form. Having got it to the form ctx^ + Qcx^y* + €y\ we can of course, if we please, make the coefficients of x* and y* unity, by writing a?* and y' for oj' V(«)) and y* *J[e). Ez. Solve the equation «* + 8aj»y - 12«*y« + l(Axy^ - 20y* = 0. We have here iSf = - 216, 7= - 756, and onr cubic is 4c> + 216tf = 756, of which c = 3 is a root. The Hessian is £r = - 6aj* + 60«*y + 72a;V + 24ay» - 636y*, ZU- ir= 9 (aj* - Aa*y - 12ajy + 82a?y» + 64/) = 9 (a:» - 2a^ - %y^\ The variables then of the canonical form are X = a? + 2y, T=.x '^Ay, which giw 6:i; = 4X+2r, 6y = X— F; whence, substituting in the given quartic, the canonical fonn is found to be 3X* + 2X*F* - F*. The roots then are given by the equations (a: + 2y)^(3)=a:-4y, (a? + 2y) 4(- 1) = a? - 4y. 204. Since J is proportional to the continued product of the X and y of the three canonical forms, and since we have just seen that the square of the product of one set of x and y is cU— -ff, where c is one of the roots of the cubic 4c' — /& + r= 0, we have J'* proportional to 4fi' -8HU''+ TU\ By calculating with the canoaical form, we find the actual value to be — c7', Or^ again, we saw in Ex. 2, Art. 202, that 16 (a/), U" H) = a'w', 16(ap,[7-fi)=aV, 16(ap3?7-ff)=aV, and in the following example that a^uvw = 32e7, hence by the values of /Oj, p,, /}, of Ex. 1 of same article 205. Prof. Cayley has given the root of the quartic in a more symmetrical form. It has been shown that apJJ-H^ ap^U—Hj apJJ-H are perfect squares severally of m, v, w. 196 APPLICATIONS TO BINABY QUANTIC9. If, farther, we enqaire under what conditiooa \u + fiv-^- vw is a perfect square, we find that V (o,c. - V) + m' («A - \) + v' [af, - V) = most be satisfied (compare Ex. 2, Art. 202), or, as it may be written, 4 — ^- — + = 0. Pt-Ps p.- Pi Pi-P, If we farther wish to make vanish with U^ we mast have X + /i* + v = 0, whence, solving, we find ^ ^ M ^ ^ * Pt-P, P8-Pi Px^P^^ thus (Pa-Ps) V(a/). Z7- 5) + 0)3- p,) V(ap. 17-5) + (p - pj \^(«P, U-H). Ex. 1. This may be verified by means of the canonical fonn, takings; for simplicity a and e = 1. If we solve the equation 4a* — a (1 + Sc^) + c — c* = 0, we find the three roots to be e, — i (c + 1), — i (c — 1) ; and the three corresponding values of E—cU are (1 - 9tf«) ^, J (3o + 1) (a;« + y«)«, | (8<: - 1) (a:«-y«)«. Now in order that any quantity of the form aa?y + /3 («« + y«) + y (»« - y*) piay be a perfect square, we must obviously have o' = 4 (/3* — y*), which is verified when a2 = l-9c«, i32 = |(3c - 1)« (3<J + 1), y* = i (3c + 1)« (3<J - 1). Ex. 2. If this methpd be applied to the example Art. 203, the other values of c are i {- 3 + 9 4(- 3)}; and the squares of the linear factors of the quartic are given in the form -2^(3){a:«-2a:y-8i^2}±i{l-4(-3)}[{l + J(-3)}a^»+{10~2^(-3)}«y-{2-10-J(-8)}s^] ±i{l + 4(-3)}[{l-4(-8)}aa+{10+2^(-3)}xy-{2+104(-8)}^]. Ex. 3. The factors of J? are the values of 0=>2 - Pz)AiflPi^U- p,H) + (/)8 - pi) 4{apt^U •- ptE) + (pi - p^^^iap^U- p^E). 206. It remains to distinguish the cases in whiph the trans- formation to the canonical form is made by a real or by an ♦ Bumside, Hermathena IV, 1876. TBANSFOBMATION, BEAL OB IMAGINABT. 197 imaginary substitution. The discriminant of the canonical form is, as we have seen (note, Art. 200), ae (ae — 9c')' ; and since the sign of the discriminant is unaffected by linear transformatioui we see that whenever the discriminant is positive, a and e of the canonical form have like signs ; and when the discriminant is negative, unlike signs. Now the form ax^ -f 6ca?y + et/^ evidently resolves itself into two factors of the form, either (aj* + Xy') (oj* + /ty") or (a;' — \y') (a;* — fiy') ; that is to say, the quartic has either four imaginary roots or four real roots. On the contrary, if a and e have opposite signs, the two factors are of the form [x* + \y') {x* — a*^'), or the quartic has two real and two imaginary roots. Hence, then, when the discriminant is negative, that is to say, when 8^ is less than 27 T', the quartic has two real roots and two imaginary; and when the discriminant is positive, it has either four real or four imaginary roots.* Now the discriminant of the equation 4tc^ - /Sb + 2"= is 27 T* - 8% therefore (Art. 167) when 8^ is less than 27 jT', the equation in c has one root real and two imaginary ; in the other case it has three real roots. Hence when a and e have opposite signs, that is, when the quartic has two real and two imaginary roots, the transformation can be effected in one way only. Next, if a and e have like signs, in which case the equation can be brought to the form a5*+ G/najy + ^*, it is easy to see that the equation can by two other linear transformations be brought to the same form; for write aj + y and x-y for x and y, and we have (1 + 3wi) a;* + 6 (1 - wi) a^y + (1 + Swi) y\ Write a;4-yV(— 1)) and aj-yV(-l) for x and y, and we have (1 + 3m) aj* + 6 (tw — 1) aj'y* + (1 + Sm) y*. Hence when a and e have the same sign, that is, when the quartic has four real or four imaginary roots, though there are three real values for c, one of these corresponds to imaginary values of x and y ; and there are only two real ways of making the transfor- mation. * The signs of the inTariants do not enable us to distinguish the case of four real roots £rom that of four imaginary; but the application of Sturm's theorem shews that (the discriminant being positive), when the roots are all real, both the quantities b^ — ae and 8aT+2 {b^ — ac) 8 are positive, while if either is negative the four roots are imaginary. (Cayley, Quarterly Journal, vol. IV., p. 10). 198 APPLICATIONS TO BINARY QUANTICS. The same thing may also be seen thus. Imagine the quartic to have been resolved into two real quadratic factors (a, J, cjar, y)\ [a\ J', cjx, y)* \ then these two factors U^ Kcan, by simultaneous transformation, be brought to the forms AX^-^BT"^ A'X^'-k-B'Y^ where Z' and Y* are the values of XZ7+ F" corresponding to the two values of X given by the equation (ac-J«)X« + (ac' + ca'-2M')X+(aV-y«) = 0. In order that the values of X should be real, we must have the eliminant of the two quadratics positive, or (a-a')(a-i8')(/3-a')(/3-/3') positive. Thus then when the quantic has four real roots, if we take for a and )3 the two greatest roots, and for a' and P the two least ; or, again, if we take for a and )3 the two extreme roots, and for ol and 6' the two mean roots, we get real values for X. In the remaining case we get imaginary values. K- either of the quadratics has imaginary roots, the resultant of the two is positive, and the values of X real. 207. Conditions for two pairs of equal roots. If any quantic have a square factor a?^ this will be also a factor in the Hessian. For the second differential U^ contains a?*, and C^, contains a?, therefore x^ will be a factor in C/Ij D^ — DJ,*. If then a quartic have two square factors, both will be factors in the Hessian, which, being of the fourth degree, can therefore differ only by a numerical factor from the quartic itself. In fact, if a quartic have two square factors, by taking these for a? and j", the quartic may be reduced to the form cx*y^ ; but, by making a, 5, d^ e all =0, the Hessian, as given Art. 202, reduces to - Sc'o^y . Thus then by expressing that a quartic differs only by a factor from its Hessian, we get the system of conditions that the quartic shall have two square factors, viz. ac — J'' __ ac?— Jc _ ae + 2hd— 3c* ^ he - cd _^ ce — d* "a 2J 6c "■ 2d e * COMPLETE SYSTEM OF THE QUARTIC. 199 a system equivalent to two conditions, as may be verified in different ways. We have, in Art. 138, given other ways of forming these conditions. From the expression (Art. 202) for the covariant J in terms of the roots, it appears that every term of it must vanish identically if any two pairs of roots become respectively equal. This also follows from the consideration that J is the Jacobian of the quartic and its Hessian, and must vanish identically when these two only differ by a factor; now the coefficients in J are, only in a different form, the conditions already written. Again, we have said. (Art. 13S) that in the same case the covariant S (a — )8)' (j3 — 7)* (7 — a)* [x — 8)* vanishes identically. But this, it will be found, is the same as 3 TV— 2SH] and we can easily verify that this covariant vanishes when the quartic has two square /actors / for, making a, i, J, ^ all = 0, U reduces to QcxYj ^ to - 3oVy' , jT to - c% and 8 to 3c*. Thus, then, we see that in the system of conditions given above, ST the common value of the fractions is --7. 208. We next show by Prof. Cayley's method (Art. 194} that the system of invariants and covariants already given is complete. We start with the semlnvariants a, ac — b*y o*rf - 3a Jc + 2 J", a'^e - 4a*bd + 6ab*c - Sd\ the first three being the leading coefficients of 27, JSj J, Since any relation between the leading coefficients of covariants implies a similar relation between the covariants themselves, there will be no incon- venience in calling the first three terms by the names 27, H^ c7; the fourth we shall call provisionally L. If now we make a = 0, we have Z7' = 0, S^' = - b% J' = 2 J', i' = - 3A* ; and by elimi- nating b between the second and third, and second and fourth of these equations, we have 4S^'* + e7'^ = 0, 3fl^"+i' = 0* Now these two quantities which vanish on the supposition a = 0, * It is easy to see that any result of elimination, obtained by combining these equations differently, will vanish when the two equations, written above, are satisfied. 200 APPLICATIONS TO BINARY QUANTICS. will, when we give H^ J^ L their general values, be divisible by a power of a. The first has been already discassed in the theory of the cubic. It gives 4J?' + J " = U^D^ where D = a^d^ + 4ac' - ^ahcd + 4 J'cZ - Wc\ The second, treated in like manner, gives 3-ff* + i/ = Z7"/S. We have thus been led to the two new seminvariants 2>, 8^ and we may dismiss L^ which we have seen can be expressed as a function of simpler covariants. Making a = again in D and 5, we have i)' = 5»(4JcZ-3c»), /S'=(-4Jrf+3e«), whence, since jEr' = — J', we have D' --H'S'^O. And giving jD, H, 8 their general values, we find i>-^/S=- ZJT, We are thus led to the new invariant T and may dismiss i>, which has been linearly expressed in terms of simpler covariants. Making a = in T, we have T' = 2bcd — eb^ — c', and we cannot now by elimination of J, c, rf, e obtain any new relation between H'j J' J 8' J T'. The system is therefore complete, consisting of Z7, Hy •/", 8y T with the equation of connection ^E^\r^V^{fl8-^VT). 209. We have already (Art. 190) mentioned Mr. Burnside's remark on the identity of the theory of the quartic with that of a pair of conies {Gonica,^ Art. 370). By the substitution a?, y, z for ^y 2iry, y", the quartic becomes ao^ + cy^ -V ez* + 2dyz + 2czx + 2 Ja;y = 0, with the identity ^xz - ^' = 0. Calling these two conies u and v the discriminant of w 4 Xv is 4\' - 5\ + 2^= 0. Thus we see that the invariants of the system of two conies are also invariants of the quartic. The solution of the quartic evidently is given by the cubic in X just written ; for if \ be one of its roots, we know that the ternary quadratic is resolvable into two factors. The discriminant of the resolving cubic, 27 T* — /8', which vanishes when two conies touch, gives also the condition that the quartics should have equal roots. To the Hessian of the quartic answers the harmonic conic [Conies^ Art. 378) of the system of two conies, and to the sextic covariant c7, which is the Jacobian of the quartic and its Hessian, answers the Jacobian INYOLUTION OF A QUABTIC AND ITS HESSIAN* 201 of the two conlcs and their harmonic conic; that is to say, the sides of the self-conjagate triangle common to the two conies. The expression for the square of c7 in terms of 27, H^ 8^ T, answers to the expression given, Conies^ Art. 388a. 210. Since JET is a cov&riant of U^ it follows that if a and $ be any constants, a 27+ 6)3J7 will be a covariant of 17, whose invariants also will be invariants of 27. The following are the values of the 5, T, and discriminant Bj of this form : 5(a27+6/3JBr) = 5a* + lgra^ + 3S*i8", B {aU+ QI3H) = 5 (a* - dSafi* - 54r^)'. The last is a perfect square, because, as was already mentioned, instead of six cases where aU-\-Ql3H has a square factor, we have three cases where it has two square factors. Hermite has noticed that if we call G the function of a, ^8, a"-9/ax/3*-54i58*, then the values just given for the 8 and T of a27-f 60HsiXQ respectively the Hessian and the cubicovariant of (7. The discriminant of G differs only by a numerical factor from the discriminant of U. The covariants of a 27+ QfiH are also co variants of U. Its jH essian is {afi8+ 9/3'T) 27+ (a' - 3^8) fi, which is the Jacobian, with respect to a, )3, of G and a 27+ BfiH. Since e7 is a combinant of the system 27, H^ the Jot a 27+ QfiH will be the same, multiplied, however, by the numerical factor G. The Hessian of J is 8' IP -SQ TUB +1283"^ which is the resultant of a 27+ B/SS and the Hessian of <?. Prof, Cayley has thrown this into the form (^8U- ^SJ + P (-S' - 27 T*) S; shewing that it is a perfect square when the discriminant of 27 vanishes. Ex. 1. For the form aU- /3ir the function G of o, /8, is 4a« - Sa^ + TfiK Thus the quartics of the system a 27— /SJT whose diacriminant vanishes are deter* mined by the reducing cubic of 27. l?he same cubic determines what quartic of the system coincides with its Hessian. Ex. 2. The factors of a £7 - /3£r are, as in Art. 205, the values of 0>2 - Pz) 4(<3o/>i - a) 4[apiU- ^) + Oo, - p^) 4(fiap^ - a) ^{ap^U - H) + G^i - f>») i^Pt - «) ^{apzU-H). DD 202 APPLICATIONS TO BINARY QUANTICS. 211. Since It has been just proved that the Hessian of the Hessian of a quartic is of the form aTU^-PSH^ we can infer, as in Art. 194, that the same is trae of the Hessian of the Hessian of any quantic For if we form the Hessian of u„M„ — M,,*, this involves the second, third, and fourth differ- entials of M. But, by the equations (»i- 3)tt,j, = an^„„+yMj„,i &c., we can express the second and third differentials in terms of the fourth, and so write the second Hessian as a function of the fourth differentials only, and of the x and y which we have introduced, and which, it will be found, enter in the fourth degree. It will then be a covariant of the quartic emanant. Now every covariant of a quartic is a function of U and H (Art. 208), and when the covariant is of the fourth degree it must be a linear function of these quantities. Actually it is found to be proportional to (2n — 5) TU^ SH^ where 8 and T are invariants of the quartic emanant and, as in Art. 141, covariants of the higher quantic. 212. System of a liquadrattc and quadratic. This system is most easily dealt with by Mr. Burnside's method. Art. 190. Let the quadratic be ox* 4- 2/8iry + 7y*, and let the quartic be given by the general equation, then (Art. 190) this is equivalent to the system of two conies and a right line ^ + cy' + ^«' + 2e?y« + 2c2ia; + 2 Jajy, 4ir« — y*, ax + jSy-^ryz^ the properties of which have been discussed. Conies^ Art. 370, &c. For example, the formula of Conies^ Art. 377, expressing the resultant of the three ternary quantics in terms of simpler invariants, gives at the same time an expression for the resultant of the two binary quantics. The formula just cited gives the resultant as <^* — 422', where 2=a*(ce-eZ') + i8"(a6-c") + 7«(ac-J") -f 2 (J)c-ad) /3y+2 [hd-c^ ya + 2[cd-be) al3^ 2' = 4(a7-/S»), <t> = 6a' + 4c/S* + 07' - 4 J)87 + 2c7a - 4rfai8. In the above, 2' is proportional to the discriminant of the quadratic, (f) is an invariant got by substituting differential symbols in the quadratic, squaring, and operating on the quartic; SYSTEM OF QUABTIC AND QUABRIC. 203 if we operate in like maimer on the Hessian of the quartic, we get an invariant of the same order as 2, but differing from it by a multiple of 82^. If we treat 2, S', ^ as conies and form their Jacobian, we get another invariant of the system of conies, the vanishing of which geometrically represents the condition that the right line shall pass through one of the vertices of the common self-conjugate triangle of the two conies. It is a*(3c&-2i'-W0 + a*7(3Jce-2W"-afe) + ay (oie + 26y - 3aci) + y (ay + 2 J* - 3a Jc) + fia* {BccT + 2 Jde - 9c*e + 06*) + affy (6ad" - 6 J'c) + i87»(-aV-6J'c + 9ac'-2aM) + ^a(12Jce-8W-4(Kfc) + /S'y (- I2acd + SVd + 4abe) + jS" [lacT - 4A*e). This is a skew invariant of the binary system of the quartic and quadratic. The formula {Conies^ Art. 388a) gives an expression for the square of this in terms of the other invariants. From what has been stated, as to the geometric meaning of the skew-invariant, it follows that if it vanishes two of the right lines which pass through the intersections of the two conies intersect on the given line ; that is to say, these equations are of the form L±M=Oj where L is the given line and M some other line. The corresponding property for the binary equations is, that the vanishing of the skew-invariant is the condition that the given quartic can be resolved into two quadratic factors L±M where L is the given and M some other quadratic The system of quartic and quadratic has only these six independent invariants now indicated ; viz. the 8 and T of the quartic, the discriminant of the quadratic, those which we have just called 2 and ^, and the skew-invariant. We get immediately two quadratic covariants of- the binary system by introducing differential symbols into the given quad- ratic, and operating on the quartic and its Hessian. Thus we get the two forms (ca - 2Ji8 + 07) ir^ + 2 (rfa - 2ci8 + J7) ary + (ea-2rfi8 + 07)^" ; {a (ae + 2M- 3<^ - 6^8 (arf- &c) + 67 [ac-h*)] of + {6a (Je-ci) - 4i8(ae+ 2W-3c') + 67 (aJ-Jc)} ocy + {6a(ce-cP)-6i8(Sc-crf) + 7(ae+2M-3c')}y'. 204 APPLICA'TIONS TO BINARY QDANTIC8. To these two binary coTariants answer covariant right lines in the ternary system, which are found by taking the pole of the given line with regard to either conic and then the polar of this point with regard to the other. Having now three quad* ratios, viz. the given one and the two just found, we obtain, three more quadratic covariants by taking the intermediate covariant (Art. 193) of each pair; and these six quadratics complete the system of quadratic covariants. There are five qnartic covariants, viz. in addition to the given quartic and its Hessian, the Jacobian of the quartic and quadratic, of the Hessian and quadratic, and of the Hessian and the first covariant quadratic. Lastly, there is the sextic covariant of the quartic. The eighteen forms enumerated make up the complete system. Ex. 1. Bequired the right lines which have the same pole with respect to each oonic. The first of the above quadratic coyariants identified with ox* + 2fixy + y^ gives ay " 2bfi + ca = Xa, 3y - 2cj3 + <fa = X/3, cy — 2dfi + ea = \y; wbencOi eliminating a, /9, y, a, 6, c-X bf c + iA, d =0, c — X, d, e the ledndng cnbic of the biquadratic (see Art. 172) is thus found : hence there are three such lines, and the reduction of a quartic to its canonical form is again seen to be the same problem as that of two conies to a self conjugate triangle. Ex. 2. The above skew invariant is also found by operating with the sextio covariant of the biquadratic on the cube of the quadrati*. Geometrically, we may determine it, either by expressing that the quadratic is a pair of a system in involu- tion with the biquadratic, or by expressing that the quadratic is harmonic with one of the factors of the covariant ^extic (Art. 202, Ex. 1). 213. System o/two.cubics. We begin with those invariants of the system of two cubics (a, J, c, d^Xj y)', (a', &', c', cnj[xj y)', which are also combinants. The simplest is (see Art. 140, Ex. 2) {ad^) — 3 (Jc'), which we shall refer to as the invariant P. The properties of this system may be studied most conveniently by throwing the equations into the form a form to which the two cubics can be reduced in an infinity of ways. For, the cubics contain four constants ^each, or eight SYSTEM OF TWO CUBICS. 205 in all. And the form just written contains six constants ex- plicitly ; and Uj v, to contain implicitly a constant each, since u stands for x + Xy^ &c. The second form then is equivalent to one with nine constants, that is to say, one constant more than is necessary to enable us to identify it with the general form. Any three binary quantics of the first degree are obviously connected by an identical relation of the form au + fiv-^yw — 0. We write a?, y, z for aw, )8r, yw^ so that the two cubics are Ax'+ j%'+ Cz% udV + JS'y' + Crz% where aj + y + « = 0. Putting for z its value, and writing the cubics {A - G,-G,-C,B-OXx,y)', {A'-C',-C\-C',B'- CJz,i,)'i then forming the invariant P of the system, we find it to be {BG') + {CA') + {AB'). The resultant of the system is found by solving between the equations ^aj'+ %'+ C«'= 0, ^V + 5'y' + GV = 0, whence we get a»=:(50'), y' = (G4')> «'=(^i5'); substituting in the identity oj + y + « = 0, the resultant is (SO')* + ( GAf + {AR)i = 0, or {(50') + (C4') + {AR)Y=^ 27 {BC) [CA') {AR). Now, if we denote the two cubics by u and v, it has been proved, Art. 180, that there is an invariant, which we shall call Q, of the third order in the coej£cients of each cubic, which expresses the condition of its being possible to determine X, so that t« + Xv shall be a perfect cube. This invariant is identical with the product [BC) ( CA') {AR), which is of the same degree in the coefficients. For, if any factor {AR) in this product vanish, Av — A'u evidently reduces to the perfect cube {AG^)z\ It follows then that the resultant is of the form P' - 27 Q. 214. If it were required to form directly the invariant Q for the form (a, 5, c, a]j[xj y)', (a', d\ c\ d'jxj y)\ we might proceed as follows. If u + Xv be a perfect cube, its three second differentials will simultaneously vanish ; or, for proper values of 206 APPLICATIONS TO BINARY QUANTICS. Xy tfj \j we have ax + hy + X [a^x + Vy) = 0, bx-k-cy-^-X (h'x + c'y) = 0, ex -{■ dy + \ (c'oj + cTy) = 0. Solving these equations linearly for Xj y^ \Xj Xy^ and then equating the product of x by \y to the product of y by Xx^ we get for the required condition a', b% a V, c', h or {J (ic') + c (ca') + A (oJ')} x {o' (of) + V (d&') + c' (Jc')} = {i' (Jc') + c' (ca') + «r (o5')l X {« K) + * W + « (*«')}» whence - Q = (ic')' + (ca')* (of) + (W)* (a6') - 3 (aJ') ( Jc') (a?) - (a<r) (Jc')' - (arf') (aJ') (of). In a di£Perent form, bj eliminating X, we have the equations aa; + Jy _ hx-\-cy _ cx+dtf a, 6, a' a', 6', J a^ 0} & J, c, y X J', c', c s J, c, c' X Cj rf, c' c'l <?, <? c, c7, c? ax + h'y Vx + dy c'x + cfy ' which are therefore {ah') X* + {ae') xy + (ic') 3^ = 0, (oc') a* + [(acf) + (ic')] a!y + (Jcf) y* = 0, ( Jc') »* + ( W) aiy + (cd') / = ; (aJ'), (oc'), (6c') (ac'), (a(f) + (Jc'), [M) = (?. (Jc'), (6^), (ci') Again, eliminating x : ^ from the original three equations we get & + X6' "■ cTx? ~ d-VXdf ' whence ac - J' + X (ac' + ca' - 266') + X" (aV - J'«) = 0, aeZ-Jc + X(acr + (?a'-Jc'-cJ')+>'"(a'^-6'c')=0, Jd-c* + X(Jcr+ (?y-2cc') + X*(yrf'- = 0, FORMS OP THE COMBINANT INVARIANTS. 207 = «; BO that ac - J', ac' '\-ca' - 2 JJ', aV - b"* ad—bcj aX + daC — Jc' — ci', a'(f - JV or yet again, eliminating dialTtlcallj, 0, 0, a, J, a', 6' 0, 0, J, c, y, c' 0, 0, c, eZ, c', (f = C. a, 5, a', J', 0, J, c, i', c', 0, c, (?, c', d\ 0, If, as in the last article, we give a, &, &c. the values ^ — 67, - 0, &c., this Q would become (SO') [GA) {AE). If then we subtract twenty-seven times this quantity from {(a(f ) — 3 (Jc')}', we get the resultant in the form J2 = (a^)» - 9 (a^)« (Jc') + 27 (ca')' («f ) + 27 (dJ')" W - 81 (aiO (5c') (cd') - 27 (ad') {aV) (cd'), a result which agrees with that of Aft. 80, it being remembered that there the cublcs were written without binomial coefficients* 215. We have, in Art. 213, formed the invariant P of the system A^ + By^ + Cz\ A'x^ + B'y^ + (7'«', by first reducing them to functions of two variables, and then calculating the value of [act) —3 (Jc'). We shall, for the sake of establishing a useful general principle, give another way of making the same calculation. We know that we may substitute in any binary d /7 quantic ;t- i "" ^ for a; and y, and so obtain an invariantlve sym- bol of operation. Now when this change is made in a function ex- pressed in terms of a;, y, z^ where ;s is — (a; + y), we must for z write -= -J- n And when the operation is performed on a ftmction similarly expressed, since its differential with respect to x will be -^ + -T- V- J or, In virtue of the relation between a?, y, z^ -1 — J" J ^^ s^® *^*^ *^® r'^^® ™*y ^^ expressed, that in any 208 APPLICATIONS TO BINARY QUANTICS. covariant we may sabstitute for Xj y^ z respectively d d a d d d dy dz^ dz dx* dx dy^ and so obtain an operative symbol which we may apply to any covariant expressed in terms of x^ y^ z^ without first reducing it to a function of two variables. Thus, in the present case, we find the invariant P by operating on ^ V + By^ + C V, with "^[dy^dz) "^^(S;"^) "^ ^V^""^)' and the result only differs by a numerical factor from the foregoing expression [BC) + ( GA') + [AR). _ In like manner we find that, in the symbolical notation, 12, as applied to a function expressed in terms of x^ y^ z^ denotes 1 dx. dx^ 1 , d_ d 1 d_ da, d or The Jacobian of the system of two cables is a combmantire coyariant, whose value is 1,1,1 Aa?, By*, C^ A'a?, Ry', Ce' {BC) /a* + ( CA') «"»« + {AR) «y. This is a quartic, for which the two invariants may be expressed in terms of the combinants which have been enamerated already. Fatting in for »', {x + yf, and multiplying the Jacobian by riz to avoid fractions, we get a = 6{GA'), S = 3(C4'), e = 6{BC% d=S{BC'), c = (BC') + { CA') + {ABT) = P, whence /S-SP*, T=54.Q-P'. We have seen that the dis- criminant of a biquadratic is iS" - 27 T*. The discriminant of the Jacobian, therefore, is proportional to ^ (P* - 27 Q), which agrees with Art. 180. C0VABIANT8 OP TWO CUBICS. 209 216. The value of the Jacobian of the cables w = or* + ibQi?y + 3ca?y* + df^, V = aV + 35 Vy + Zc'xjf^ + cf y, is {ab') x^ + 2 (oc') a^y + [(oc?') + 3 (J^] ^y + 2 (i(f ) ipy»+ (aT) y* ; writing this J= a^a* + 4ajaj'y + 6a,a;y + 4a,a:y' + a^*, we have, since, P= (oe?) — 3 (Jc'), {«&') = «•? K) = 2a„ (atf) = 3a, + iP, (Jc')=«,"-i^i (&cf) = 2a„ (ccr)=:a^. Accordingly, the identity (be') [ad') + [ca^bfT) + (oJ') (ctT) = gives us a^a^ - 4ka^a^ + 3a/ = ^^^P" ; also substituting these values in the determinant form of Q expressed by means of them, p. 206, we find The covariant 12' of ti and v is, in full, (ac' -[-ca'- 2W)»* + (a^+ ^ - 5c'- cJ') xyi-{bct+db'- 2cc')y\ which we write = fl^ = ttjaj* + 2^jajy + 7^^*, thus oo' + ca' - 255' = a„ ac?' + c5'-25c' = i8j + iP, 5c' + da'-2c5' = i8j-iP, bi ^-db' -2cti ^^^. If the Hessians of u and v be written respectively E^ oa?" + 2^icy + 7y*, -ff ' = ot'ic' + 2/8'ajy + Vy', where ac - 5' = a, &c,, we find, as in Ex. 6, p. 24, 8 (ay + 7a' - 2^8/3') = 4 (a,7, - /S,') + P". The results of operating with either cubic on the Jacobian are two linear covariants, which may be compared with the results of operating on the cubics with the Hessians ; it is easy to see that we have thus the different ways of writing them, oc' - 2^85' + 7a' = - (a7i - 25/9, + ca,) = J (a% - 35a, + Zca^ - da^ = - { J (5c') + c [col) + d [aV)\ = i„ oef - 2i8c' + 75' = &c. = i,, ca' - 25^ + a7' = &c. = L\^ dd - 20/8' + 67' = &c. = i'„ 210 APPLICATIONS TO BINARY QUANTICS. denoting these two linear covariants by i = ijX + i^, L' = L\x + 1/'^, or In determinant form a'«+ J'jf, a, h i = i'a? + c'y, J, c , i' = c'a: + (fy, c, cf It Ib obvious that their determinant a, 2^, 7 ax + by J a', i' cx + dtfj c', <f a', 2/3', y = <2. Ex. For the forms in Art 215, the above qnadiicoyaiiants are BC$fz + CAzx + ABxy, {BC + CB^yz+ {CA' + AC) zx + {AB' + 5^') ay, B^C'tfz + C'i4'«« + il'5'ay, and the above linear covaiiants are found by operating by the method of that article to be A'BCx + B'CAy + C'ABz and AB'C'x + BC'A'y + CA'B'z. 217. There Is another form in which the system of two cublcB may be usefully discussed, viz. ox' 4- ibx^y + Zcxy* + rf/, hx^ + Scx'y + Zdocy^ + ey'. In other words, the cubics may be so transformed as to become the differential coefficients of the same quartic, with regard to x and y respectively. We can Infer, from counting the constants, that the proposed form is sufficiently geperal ; but the possibility of the transformation will be more clearly seen If we consider the two linear covariants just obtained : if we make a', &', d^ c? = J, c, c?, e, we have U = Tx^ L = Ty. Thus we see that, in order to effect the proposed transformation, we are to take these two linear covariants for the new variables. If u and V be the differentials with regard to x and y of the same quartic, the quartic itself can only differ by a numerical factor from xu-\-yv] and in fact L'u-^ Lv is immediately seea to be a combinant, as being a function of the determinants {aV) &c. The leading term in Uu 4- Lv is {aV) {be') + [ab') [ad') - {acj. INVARIANTS OP TWO CUBICS. 211 "Whence, sabstituting the values given in last article this leading term is found to be 4 {a^a^ - a^) + J Pa^. But ajx^ - a^ i^ the source of the Hessian of the Jacobian ; accordingly nH[J) -h PJ=^ 3 {L'u + Lv). Thus, in order to obtain the quartic, having for its differentials the two given cubics, we must add twelve times the Hessian of the Jacobian to the Jacobian itself multiplied by P. For the system of cubics considered in this article, / is the Hessian form of the quartic, P=rte-4&c?4-3c', . a, i, c and C = i, 0, d c, d^ e 218. If we have any invariant of a single quantic, and if we perform on it the operation «' 3~ + ^' jI + &C'> we obtain an invariant of the system of two quantics of the same degree. For if we form the corresponding invariant of w + Xv, the coeflScients of the different powers of X will evidently be invariants of the system. Thus let the discriminant of w + Xw be i> + 4Xilf + 6XW+ ^yJM! + X*i>', and we have the three new invariants if, N^ M\ whose orders in the coefficients are (3, 1), (2, 2), (1,3). If in general we form any invariant of t^ + Xv, and then form any invariant of this again considered as a function of A| the result will be a combinant of the system u, v ; that is to say, it will not be altered if we substitute lu + ww, Vu 4- w'l; for w, V, For, by this substitution, we get the corresponding invariant of (Z + \V) w + (wi + \wl) v, which is equivalent to a linear transformation of X, by which the invariants of the function of X will not be altered. If then it be required to calculate the invariants of the biquadratic, which we have found for the discriminant in the case of two cubics, we may, without loss of generality, take instead of u and v two quantics of the system w + Xi? which have square factors, taking x and y for these factors ; and so write u = a^ + ibx^y^ v = 3cxjf* + dy*. For this system we have P=^ad- Bbc^ Q^V(? {ad-^bc)\ the 212 APPLICATI0S8 TO BINABT QUAHTICB, resultant F*-2TQ being a*d' {ad - 9hc). Now, for this form, the biquadratic is 4acV + (aV - 6a5crf - 36V) X' + 4rf5 Diiiltiplying by aix to avoid fractions i> = Z>' = 0, M M' = 6ac', N= a'd' - &abcd - 3JV = P'- 126V. Hence, T= 2NMM' - N^ = -{P^-S6P'Q + 216 Q^, whence the discriminant of the biquadratic S° — 27 2™ is pro- portional to ^ (P*- 27 Q), which agrees with Art. 183. The method naed above is evidently also applicable to covarianta. Thus let the Heasian of u + \v be E+ XH^ + \'H\ and we are led to the intermediate quadratic covariant M„ whose leader ia ac +ca' — 2bb'. The covarianta and invariants of the BjBtem of three quadratics, just mentioned, are also covariants and invariants of the system of two cubics. Thus if we take the Jacobian of each pair of quadratics, we have three quadric covariants of the orders (3, 1), (2, 2), (1, 3). We have seen (Art. 167J that a cubic and its Hessian have tlie same di^ criminant, and therefore we may identify the discriminant of M+ \II^ 4 Vfi"' with the expression already found for the discriminant of u + \v. Now if m, w, ic be three quadratics, the discriminant of Xu + fiV-\- vw is plainly X'Z*,, + 2X^i>,j + &c. Thua for the system under consideration, we see that i>,j, D^, can only differ by a factor from M, M' already enumerated; and we have the two invariants 2),^, 2i?y whose sum similarly is, to a factor, the same with N. Another relation was already found (Art. 21G) between them and P. These invariants, there- fore arc not new but can be expressed each in terms of N and the combinant P. The expression is most easily arrived at by taking the particular ease already considered u = ax' + Sbs^tf, V = 3cxT/^ ■+ d^'j in which case we have H^=aca^+[ad-hc)x)/-\-ddf/', 2Z),3=6V, iD^=%abod~a'd'-h'(?. Thua we find, for the case when the discriminants of both cubics vanish, the relations P'-JV=242>,„ P' + 2iV=- I2i>„; and it can easily be verified that these relations are true in Gdb', pro- I Lastly, we may form the invariant M of the system of three quadratics, but we have found this already to be the combinant Q,. COMPLETE SYSTEM OF TWO CUBICS. 213 219. We are able now to give a list of the covariants of the system, which the investigations of Clebsch and Gordan show to be complete.* There is one qtmrtic covariant (1, 1), the Jacobian (Art. 216). There are six ctdnc covariants^ the two cubics themselves (1, 0) (0, 1), their two cubicovariants (3, 0) (0, 3), and the two Jacobians (2, 1) (1, 2) of either cubic combined with the Hessian of the other. For the canonical form, the last four covariants are included in the form 1,1,1 Ax* , -By" , Cfe" A{By+Cz)j B{Cz'\-Ax)y C{Ax + Btf) according as we accentuate the coefficients in neither, either, or both of the last two rows. There are six qtLadratic covor riantSj viz. the two Hessians (2, 0), (0, 2), and the intermediate covariant (1, 1), these three being for the canonical form 25(7y«, 25' (7>, ^{BO' + B'G)yzi and for the remaining three covariants (3, 1), (2, 2), (1, 3), we may take either the Jacobians of each pair of these (Art. 218), or the results obtained by operating with each on the quartic covariant. There are six linear covariants, viz. the two (1, 2), (2, 1) considered Art. 216 ; two (3, 2), (2, 3) obtained by operating with the Hessian of either cubic on the cubicovariant of the other; two (1, 4), (4, 1) obtained by operating with either cubic on the square of the Hessian of the other. Lastly^ there are seven invariants, viz. the two discriminants (4, 0), (0, 4), the combinants P, Q, (1, 1), (3, 3) (Art. 213), and the invariants Jf, M\ Nj of Art. 218. Of the preceding invariants Pand Q are skew. We have in Art. 218 connected P* with the in- variants 2)j3, D^ of that article, and the expressions there given for the S and T of the biquadratic are, in fact, expres- sions for PQ and Q* in terms of N, if', &c. We can also con- nect Q* with the functions 2)^, jDj,, &c. if we remember that, as was remarked (Art. 218), Q is the invariant 12.23.31 of a * Of the eight linear coyariants which they supposed to be inedndble, Sylvester has shown that two are not, and d'Ovidio and Gerbaldi have developed their expressions^ Aui di ToriiMf xv. 267. 214 APPLICATIONS TO BINARY QUANTICS. system of three quadratics, and that it was proved at the end of Art. 193 that double the square of this invariant is -Ou, ^„, A. /y^,, //,„ Jj^ A., -D„, ^„ 219a. Geometrically, the two cubics may be taken as two triads of points on a conic, as in the case of Art. 198. When any point {x'y') on the conic is joined to the points of the triad [u\ the joining lines meet the sides of the triangle {u) in points forming a triangle in projection with [u\ and it is easily seen that the first emanant with regard to x\ y' of the cubic u^ whose roots are represented by the points of the triad, is the axis of the projection. Similarly, may be made the centre of a projection for ?;, and the axis corresponding be found ; these two axes intersect in a point 0\ and as moves round its conic^ O describes another conic which intersects the conic in four points representing the roots of the Jacobian of u and v. The four corresponding points can without difficulty be de* termined. The roots of the covariant H^ are represented by two points on the original conic whose axes with respect to the two triads are conjugate lines with respect to that conic. In Art. 198 it was seen that for a system of a cubic u and a quadric V the simplest linear covariant is the centre of the projection with regard to u whose axis passes through the pole of F. If we calculate for the triad consisting of this centre and the two points where V meets the conic, the combinant P with the triad m, we find P=0. It follows from symmetry that each of the vertices of the former triangle is the centre of a projection with regard to u^ of which the axis is conjugate to the opposite side ; and that such relation holds mutually between u and V when P=0. Another way of stating the condition P= is that the triads are such that H and its pole are axis and centre of a projection with regard to i?, and H! and its pole, axis and centre of a projection with regard to u. The simplest linear covariants discussed above are represented as follows : L is the centre of the projection with respect to r, ANY SYSTEM OF BINARY CUBICS. 215 whose axis is the line joining the poles of H^ H.' ; and i' the centre of the projection with respect to u of the same line. The identities established, Art. 216, show that L is also centre of the projection with respect to w, whose axis joins the poles of if, S^^ and Z' centre of the projection with respect to v, whose axis joins the poles of H.^^ H\ Thus the pole of jff, can be constructed. If the points i, L' coincide, JB^ -H,, H^ are concurrent lines^ and the line joining their poles is the axis of projection for the L point with respect to both triads and the combinant Q vanishes. 219J. System of four cuhtcs. It is desirable to generalise to this case the theorems given p. 181 for three quadrics. For the determinant of the system, if we denote the combinant (a^e?,) — 3 (JjC,) by P^,, &c., we have, by Art. 33, Ex. 4, the relation P^Pu + P,rP» + -P. A = - 3 («,5M). and this becomes when a^, &c., are replaced by y", &c. = 3 «i> «,» «»> y *., K *3, -xy Cl» «.. c,. ^y c?„ d,. ^., -a;' = -3 c^x^d^^ c^-^djf^ cjc^tdjf This is the cubic which determines the values of x'^ y\ so that XjWj + X,w, + XgWg = {xy' - oiy)^ be a perfect cube, see Curves (Art. 216a &c.], and the discussion of unicursal cubics there giv^en furnishes an additional geometrical illustration of this theory* For four cubics, denoting the function just written by J^^j we hsLVQ the equations P^U, H-P«t/3 + P,3t*, = -3e/3,,, P^U^ + ^41^2 + ^2^4 = - 3e/,j„ ^28^1 + -^81^2 + ^12^8 = - 3«^128> a linear transformation in four variables* 216 APPUCATIONS TO BINARY QUANTICS. If Jj, Signify the Jacobian of the cables 0^05'+ &c., a^'+&c., we have six such Jacobians each a qaartio in Xj y^ which we may write Now it is easily foondy as in Exs. 3, 4, Art. 33, that = -4{5X + M4. + M«+Ml, + *4.^i3 + MM} = 12 (c,,c^ + c„c^ + c,,cj. Hence, and from the vanishing combinations, ^12*84 + ^84^1, + &C. = 0, &C., the linear relation had by dialytic elimination of x^^ x^jfj &c., between the six functions «/„ is found to be -Pl,«^.* + ^i,^4. + ^i/» + -P.4J'l2 + ^yiS + ^./l4 = 0. Other relations can be established connecting the functions H with these, as at p. 181, on which we need not delay. For instance, we might show for five cubics, with an obvious notation, «*4> -^141 ^U Also for four that J^J^^ + J^^J^^ + J,/^ = 0.* 219o. The system of four cubics furnishes another geome- trical view of binary cubic systems. For, we can by linear combinations alter the functions to x', aj*y, {cy\ y^ and introduce these as new variables a;, y^ Zj Wj the coordinates of a point in space which depends on a single parameter t The relations between them, and f, may be written x : y = y : z = z : to=*ti thus the point t is restricted to a twisted cubic, and the roots of any binary cubic are the parameters of the points of intersection of a plane with the twisted cubic (compare Surfaces^ Art. 337 et seq). For any point t on the curve the osculating plane is x-dyt-h3zf-wf=^0j so that the points whose osculating planes meet in any point ♦ Dr. F. Lindemann's paper {Math. Annale% vol. xxiii. p. Ill) on the geometric exposition of binary forms includes the formulas of this article. ILLUSTRATION OP TWISTED CUBIC. 217 oiy'z'w are determined by aj' — 3y'^4 3/<* — wY = 0, or the points of contact lie in the plane vSx - Zz'y + ^y'z — x^\jo — 0, but this plane passes through oiy'z'uf. Hence, in any plane F^ ax + Zby + Scz + (Zto = there is a point whose coordinates are d^ — c, 5, — a, which is the intersection of osculating planes at the points in which it meets the curve, and the point and plane may be said to correspond. When lies in the plane F corresponding to a point (7, the above relation shows that (/ lies in the plane ^corresponding to 0] and the line 0(y possesses an invariant relation to the curve. For two such planes F^ F the P invariant vanishes. Two equations of the line meeting the curve in the points f„ t^ are x-yit^-^tt^-k-ztf^^^^ y-^C^ + O + ^^^a'^^) '"d y^^ express that this line passes through the point 0, the parameters \^ t^ of the points in which the chord of the curve through meets the curve are the roots of the quadratic thus the Hessian of the binary cubic is represented by the intersections of the chord through with the curve. The determinant, whose constituents are the coordinates of the coUinear points 0, ^j, ^,, and of any other point, vanishes identically. Hence, by the first formula of last Article, if we put oajj + 3 Jy^ + 3c«, + dw^ = \y aa + 3Jy, + 3c«, + dw^ = \^ we have identically = (ajjU?, - 3y^z^ + Zzj/^ - w^x^) {ax + 3by + 3cz + dw). Hence the plane F contains the line of intersection of the osculating planes at the Hessian points. The plane through the intersection of the osculating planes which is harmonic to the former is easily found to represent the cubicovariant, and having the same Hessian, its corresponding PF 218 APPLICATIONS TO BINARY QUANTICS. point is the harmonic conjugate of with regard to the Hessian points. 219(2. A second plane 4> gives rise to a similar system, and besides to the results of combining the two. The coordinates of the tangent line to the curve at any point are ^, — 2^', ^ ,3^', 2^, 1, whence we find that Hs a root of the Jacobian of F^ ^ for any point on the curve whose tangent meets the line in which the planes intersect. The line joining OCX meets the tangents at the same points. The parameters of the points in which a chord through any point on 0(7 meets the curve are given by V-ff + 2X/X-S; + fi^E' = 0. From any point given on the curve, two chords can be drawn to meet 0(7, and if -3^ = 0, these chords meet 0(7 in points harmonically conjugate to and (7. The simplest linear co variants are represented by the point where the plane through containing the chord through (7 meets the curve again, and that in which the plane through (y containing the chord through meets the curve again. The invariant P vanishes for two planes if the point corresponding to one lie in the other, in which case the line F^ becomes identical with its corresponding line 0(7. The combinant Q vanishes when it is possible to draw an osculating plane through the line jP4>, in this case it is easy to see the corresponding line 0(7 meets the curve and the two points which represent the simplest linear covariants coincide.* 219e. System of a quarttc and cubic. This system consists of sixty-one formsf: 1 sextic, 2 quintics, 5 quartics, 8 cubics, 10 quadrics, 15 linear, and 20 invariant functions. An invariant of the third degree in the coefficients of the quartic, and of the second in those of the cubic, can at once be written * See a paper on this geometrical representation by Mr. W. R. W. Roberts, Proc, Land. Math. Society^ vol. xill. Also Dr. Lindemann's paper, Note p. 216. t In Dr. Gundelfinger's inaugural dissertation the number was assigned as 64, but Dr. Sylvester has since shown {Comptes Bendus, t. LXXXVII.) that two of the quadratic and one of the linear covariants are reducible. STSTEH OF CUBIC j^ND QUABTIC. 219 down from the determinant form of the sextic covariant, p. 192, being the result of operating -with that covariant on the square of the cubic, a -3J', 0, o, a, -2J, -3c 3c', - <r, - 35', 3c', - d' — Sbj 3c, — d Sdj - e, The same determinant with only one row so altered is a covariant cubic, and another of the same degree in the quartic coefficients may be got from the skew invariant, Art. 219. The two simplest linear covariants are a' [dx + ey) - ZV [ex + dy) + 3c' {hx + cy) - rf' (aa; + Jy), and the corresponding one for the Hessian of the quartic. The determinant just written is the eliminant of these two. Operating with the quartic on the square of the Hessian of the cubic, gives an invariant of the first degree in the coeffi- cients, of the former. 220. System of two quartics. We consider here chiefly the invariants of this system, which are also combinants. We have seen. Art. 218, that the invariants of any invariant of \u + fiv are combinants. Let us form the 8 and T of \u-\-ii,v and write them 8\^ + 2X/A + iSy , r\' + aV + «' V + Tij!" ; then if we form (Art. 198) the invariants and covariants of this cubic and quadratic, we get combinants of the system of two quartics. Thus the discriminant of the quadratic is 2' — 4,88' = a, 45, 6c, 4e^, e a\ 4&', 6c', 4^?, c' c', -rf', c, -£?, c', -J', a' c, — J, a = (ae')»+ 16(M')*+ 12(ac')(c6')-48(Jc')(ctf )- 8(ai') (£?e')-8 (a(r)(Je'), which we shall refer to as the combinant A. It will be convenient to use the abbreviations (aJ') = a, (efe')=a', (ai') = /3, (Je')=i8', [ao')^\, (cc'} = V, (Jc') = /., (<j(f ) = /a', (ae') = 7, ^d') = S. We have then A = 12\V - 48/A/x' + 7" -H68» - 8aa' - 8/8^8'. We can find by a different process an independent combinant of the same order in the coefficients. The Jacobian of the two 220 APPLICATIONS TO BINAKT QUANTICS. quartlcfl, J^ is ax'+3Xa;V+(3/8+6/x)a;y+(7+8S)a;y+(3i8'+6/)ay+3\'ay*+a^^^^ Again a combinant quadratic P or 12' is (iS - 3ytt) a?' + (7 - 28) a?y + (/S' - 3/^0 y". The sextic J has an invariant of the second order in its coefficients (Art. 141), and the discriminant of P is of like order, neither being identical with A. The condition that u and v be derived functions of a quintic, is the vanishing o( B =^ a, 5, c, d 5, c, df, e a J b ^ c ^ d! hence, or as in Art. 219J, we find P=W'-/L6/A'-/8/L6'-i8'yttH-8'-aa'; the invariant of J will be found to be ^ + 48£, and the dis- criminant of P to be .^ - 12^. The resultant of u, v, found bj expanding the determinant of Art. 84 is B = 1296\'^'* - 3456 (a/^X'" + a'/ X') - 1 1 52 (a/3X'* + o'iS'X") - 727'XX' - 5767SXX' + 9216aa>/ + 967 ()3*X' + /S^X) + 2887 (ai8'X'+ a'/SX) +1536S(a/S'/+a'/8/A) +3072aa' (i8/+i8'/t) 4 7* - 48aaV - 16i8i8'7" - 256aa'78 + 512aV - 256 (a/S^ + a'/S") - 4096aa'S*. In terms of the preceding combinants can be expressed the combinant which we have called 37 (Art. 182), but which, in order to avoid confusion, we shall now call 6^; and which expresses the condition that a quartic of the system u-\-\v can have two square factors. Such a combinant must vanish if v reduce to the single term cVy* . In such a case a, a', /8, ^, 7, 8 all vanish ; and we have A = 12XX' - 48/^^/, jB = XX'-./a/, 5 = 1296X*X'*; hence we see that [A — 48jB)' — iS is a combinant which vanishes on this supposition. And since it is of the same order that we have seen. Art. 182, that T must be, it is identical with it Using the values already given for -4, jB, -B, we find that {A - 485)" -- 5 = 128 C, where (7=-277(X/*-fXV)+18(/3V'+/3V)+188*XX'+367S/i/A-36aa>/ + I87SXX' - 97V/ - 37 (a/S'X' + a'/3X) - 248* (/8/u,' + ffyi) - 68 (/3V + /S'^'X) - 68 (a/SV + a'/8X) + 2 {d^ + a'/3') + aaV + 4aV' - 2aa'7S + 478' + 16aa'8' + 8S\ INVABIANTS OF TWO QUARTICS. 221 Again, if we form the invariant which we called / (Art. 198) of the quadratic and cabic at the beginning of this article, it will be found that {A + 485) {A - UB) - 5 = - 1287, a formula obtained by Mr. Warren, Quarterly Journal^ Vol. VII., p. 70. 221. We consider next D, the resultant of the cubic and quadratic, and E the discriminant of the cubic. D is the invariant which we have called 8 (Art. 182). It may be mentioned, that besides the methods already indicated for calculating that invariant in general, the following may be used. It is required to find the condition that \ can be determined 80 that the three expressions w,j + Xv,^, te,j + Xt?j„ u^ -f \v^ can be made to vanish together. Now we may multiply each of these by the 2 (w — 2) terms a?**"*, &c., of a quantic of the degree 2n — 5, and so obtain 6(n — 2) equations, from which we can eliminate dialytically the 6 (n — 2) quantities aj®**^, &c., \a5'*^,&c., and so obtain 8 in the form of a determinant. In the case of the system of two quartics, 2) = - 16\'X''+48\»\'V/+ 6\\>V"+ 16/iAV+ 27\V*+ 27\V 4 SQolK/jX^ + 36a'\V\' + 12V\'* {13// + ^/i) - 96a/iA'\V - 96ay\V - 67V/A" - 67/x'\V - 36SXV/A'* - 368\/x'X'' - 48a/A/A'* - 4.8a: fiy - 24.SXfifi'^ - 24SX>y + 2idl3\\'* + 24a'i8' W - 18ai8'\V - 18/3a'XV - 6SV\'»+ 877W - W'fifi' (162aa' + 90/3/3') - 36a/3/A'*- 36a'i8'/iA* + 96S*\\>/ia' + (228aa' - ^0/3/3") fji^y - 4a»7^''' - 4a'"7\' - 16ai8«\V - 16a')8"\V - 30aa''\V - 30aV\> - SOoa'XV (^8/ + ^S'/t) - 20SXX' (X/Sa' + X'a/3') + 2yfAfjk' (X/Sa' + X^a^S') + 488 V^" + 48S'aVX"+ 24a/3^/A/"+ 24a'/3 W+ SGa/SiSV + b^a^P^/jJ" + 240aa'/8/tA/iA'"+ 240aa>Vfl' + 32a8 V +32a'8*/A» + 3X»^a'' + 3XV/8'* + 24aa' ()8>'» + /3 V) - ^aa'yW - 12aVXX' - SOaa:/30'\\' + GOaa'XX'S" + 12/3/3'\\'S' + {84a*a'"+ 120aa'i8/8'- 12i8*/3'"} /*/- 192aa'S*/x/iA'+487SV/i' - 48S*XX' + Qyaor/3\ 4 ^r^o^a'ffX' + 48a'a'" (^8/ + i8>) + 24aa'S ()8y+ /S'V) - 96aa'S» (/S/i'-l- ^8'/^) - a^aV- 4aa'"i8' - 4aV/3"+ 8a'a''- 47SaV - 48aVS^- 16S*aa'/3/3'+ 64aa'8\ 222 APPUCATIONS TO BINARY QUANTICS. 222. In studying the relations of these combinants, we may, without loss of generaUty, soppose one qoartic to want the first two terms, and the other the last two ; that is, we may write u = CUD* + 4:bz*y + Scai^y\ t; = 6c'a5y+4kiijy'+ey*. To save room, we write ae = lj hd^m^ cc' ^n^ caJP-\'C^eb*^]f, We find then A^ (Z - 4tmy + 12w [I - 4»i), B^ P (Z- 16»i) + 96?^ - 72wP (?+ 8m) + 1296?n', I =- Pm'+ 4Zw'+/(Z"- 2Zm-8m") + 6y - n (P + 6Z?w" - 16wi') - 9n^' + 12n" [r-^-lm- 2^*), D^-n^ {9Zm» (Z - 4w) - 6p" (2Z'- 6Zm - 47n») - 27/+ 16^ (Z- w)'}, E^-'Vm*^ 2lmY (Z + 2?n) - (Z + 2w)*y - 2nZw' (Z + 2?w)" + 4/ + 2np» (Z + 2m)' - 18n;?Vm" - w» (Z + 2m)* + 36?i7m' (Z + 2m) - 6np* (Z + 2m) - 6ny (Z + 2m)« + 27wy + 4w» (Z + 2m)' - 108w'Zm". By the help of these values we can verify the equation 165'-^5«-2ZB+^=i), which expresses E in terms of invariants already found. The Jacobian, with this form, wants the extreme terms. There is no difficulty, therefore, in calculating its discriminant, and thus verifying the theorem of Art. 1 80. Finally, we have seen that a cubic and quadratic have a skew invariant M, The equation of connection given at the end of Art. 198, when applied to the case considered in this article, shows that the system of two quartics has a skew invariant M of the 9*^ order in the coefficients of each, whose square is given by the formula M^^A {AE-- ry - SD [r - 9^12;+ 5WE). Mr. Burnside's method, reducing the theory of two quartics to that of three conies, discussed Conies^ Art. 388, would have led us to the same results. SYSTEM OF TWO QUARTICS. 223 223. I have also sometimes found it convenient to suppose each quartic to be the sum of two fourth powers, so that for each the invariant T vanishes. Let the quartics be aw* + Jt?\ a'tr* + JV, where u is a,a; H- ^^^ &c. We use (12) to denote ttjiS, — a^/Sj, and employ the abbreviations (12) (34) = L, (13) (42) = M, (14) (23) = JV; where it will be observed that we have identically ii+J/+JV=0» No^ the invariant 8 is got by substituting -r , — -r-, for a:, y in the quartic and operating on it with the result. If we operate in this way with u upon u the result vanishes ; but if we operate on v the result is (12). We find then at once that the 8 of \?7+/AFis X»aJ(12)*+X/^{aa'(13)*+aJ'(14)*+Ja'(23)*+JJ'(24)*}+/AVJ'(34)*. The combinant then which we have called A is {aa' (13)* + aV (1 4)* + ha! (23)* + IV (24)*}« - 4.ala'VL\ In the same case B is found to be — aha!b'UMN. The invariant T is found by operating on a quartic with its Hessian. But here the Hessian of U is ab (12)'wV. We find then that the Tof XZ7+/AFi8 XV [aha: (12)^ (13)" (23)» + abV (12)» (14)« (24)«} + X/t» {a'Va (13)« (14)* (34)» + a'Vb (23)" (24)» (34)*}. Hence, we have immediately ^=-a'JV*J'*i*{aa'JV*(13)*+aJ'ir(14)*+Ja'ilf»(23)*+JJ'i^(24)*}», i?=-a'^JV*6"i« {a^a'^N^ (13)* + d^b'^M^ (14)« + Va'^M^ (23)* + Vb^'N^ (24)* - ^MNa^a'V (13)* (14)* - ^MNVa'V (23)* (24)* - 2MNar'ab[nY (23)* - ^MM'^ab (14)* (24)* + 2M'N'aba'b' {M' + N'- 2L% I =^^aba%'L\a^a'^N\nf^a%'^M^[UY^Va'^M'{2^Y^^^ + {M^ + N'^ 2U) [a^a'V (13)* (14)* 4 VaV (23)* (24)* + a'^ab (13)* (23)* + Vab (14)* (24)*} + 2iPN^ [3P + N'-- 4i«) aha'V], by the help of which values we can verify the equation already obtained. 224 APPLICATIONS TO BINARY QUANTICS. Sylvester has reduced by two the total number of independent forms for the system of two quartics which G-ordan enumerated as thirty, viz. there are for each qaartic the five forms u, H^ J^ 8^ T\ twelve more forms are got by taking the operations 12, 12', 12', 12* performed on the pairs m, m' ; «, -ff' ; m', S"; two more by 12', 12* performed on H^ H' } and lastly, four more are obtained by operating on the sextic covariant of either quartic with the other quartic or with its Hessian. This gives in all eight invariants, eight quadratic, seven quartic, and five sextic covariants. The following are examples added in further application of these principles to binary quantics of the first four degrees. In dealing with the sextic we shall return to the system of two quartics. Ex. 1. If three quadratics be the three second derived fonctionB of a single biquadratic function, express the relation between their coefficients. With the notation of p. 181, «=a|xH-2friary+^iy2, v=a^x?-{^c^ 2i)r*=arCrHx«Cf^2Mi» we must haye a„ d„ a» d, ^i» ^u ^» ^2 a„ *» <hi h ^» ^» \i ^8 This may be found by comparing coefficients when we effect a linear transfor- mation on the quadratics. It may be found more rapidly in another form, as follows : The general conic identity between the quadrics (p. 181) for n = ax^ + 2bxtf + cy^, v = baiy^ + 2cxy + dy\ w=ca? + 2dxy + ey«, has Di^ = ac-b:', L^i^bd-d^, D^-ce-d^, 2D„ = be-cdf 2i),i = ae + c^-- 26rf, 2i>,2 = ad-be, whence, by the identity {ac -b^{ce-'d^ + (ad - be) {ed - ie) + (oe - W) (W - c») = 0, = 0. deriyed from the matrix we get a, 5, Cf d bf Cf df e D,^D„ - 4D,^^ + D„ (2)„ + 22)„) = 0. Ex. 2. To give a geometrical signification of the preceding ration ? The equation {ax + by + cz) x: + {bx + ep + dsi) y' + {ex + dy + et) gl' = is that of the polar of (a;', y*, a') P to the conic u = ax^+ cy^+ez^+*2dyz+2czx+ 2bxy=0 j and if y'* = 42V, P moves on a conic » = 0, and its polar touches {ax + by + ez) {ex + dy + ez) =:{bx + cy + dzf. This conic is found to be the harmonic conic of u and v, and to meet 9 in the points which are the roots of the Hessian of u as a binary quartic. A Innary transformation leaves v of the same form as before. The three quadratics correspond of course to right lines, and if a binary transformation make them the derived functions of a binary quartic u, they must be two tangents and their chord of contact to the harmonic conic, which is the reciprocal of v with respect to u. That the reducing cubic of the quartic is clear of its second term is {Conxcsy Art. 381, Ex. 1) equivalent to this special determination of the harmonic conic F. The conies « and are said to be apolar {Apolaritcit und rationale Curverif F. Meyer, Tubingen, 1883). EXAMPLES. 225 Ex. 3. What relation holds among the coefficients of four quadratics if by the same linear transformation two of them become derived fmictions of one cable and the other two of another ? = Aii>44 -f ^«^M - 2 {2>i^« + At^« - I>iJ>Jif a„ *,, Oj, dj *i> ^l» *2> ^2 «3> *8» «4> *4 ^8> ^8> ^4» *^4 the notation being the same as in Example 1, are easily foxmd to be equal, and the condition required is got by equating them to zero. Geometrically, the lines are such that a quadrilateral can be inscribed in the fundamental conic, having its sides each through the pole of one of the given lines ; thus, points P and It can be found on the conic, such that when P is joined to the pole of line 2 and R joined to the pole of line 1 the lines meet in Q on the curve ; and when P is joined to the pole of line 4 and R to the pole of line 8 these lines must meet in jS^ on the curve. Similarly considered, Ex. 1 has the lines so placed that a quadrilateral can be inscribed in the conic, having two alternate sides through the pole of one given line and the other sides severally through the poles of the other two given lines. Ex. 4. What relation holds among the coefficients of three quadratics which can be the simplest quadric covariants ZT, Hi^ H' of two cubics ? In Art. 219 it was pointed out that P\ PCl^ Q^ can be independently expressed by means of the functions D, The values are in the present notation P« = 4 (4J9,8 - 2>22) PQ = ^ {2>„2>« + As' - Ai^M - Aa^a}, -^I2> -^2» ^ta '18» -^Vii -^Vi whence, comparing values of P^Q^ also writing the minors of the last determinant A = JDiiDzz - Duf Ac, we find The form of this relation, compared with that in Ex. 1, shows that the Jacobians of Ej Sif H' are second derived functions of a quartic. Ex. 5. A cubic can be uniquely determined. whose covariant .&| with a given cubic vanishes identically. Ex. 6. The P invariant of n = (x- ay) {x- ^y) {x^ yy), v = A {x - ay)* + B {x - ^y)* + C {x - yy); vanishes identically. Similarly, 12'* vanishes identically for two quantics u = {x- a,y) (x - a^),„{x - any)^ v = Ai{x- a,y)»+...+ An{x- a„y)*. Two binary quantics for which 12 * vanishes are said to be qpolar (see Ex. 2] : alsO} conjugate binary forms by Dr. Schlesinger, Math, Annalen XXII. Ex. 7. For two cubics, show that, in terms of the roots, (a-a')(p-^')(y-y') + («-^')(^-y')(y-a') + («-y')(^-«')(y-/3') 3P = (a-a')(^-y')(y-^)+(^-/3')(y-a-)(a-y') + (y^y')(a~/30(^-a') = aa' Ex. 8. In the system of cubic and quadratic (Art. 198), calling the Jacobian of U and F, v, prove that the combinant P for th^ cubics u, U, is double the inter* mediate invariant /, and that the P for the cubics u, /, vanishea. aa 216 APPLICATIONS TO BINARY QUANTIC8. Ex. 9. FoT the cabiooYaiiantB /, /' of two cnbics «, v the combinant P (7, /O = 4i),^ - 2Q. &L, 10. CUcalatmg, as in Art. 216, the determinant form = i2 of the resnltant of two cnbics (Ex. 2, Art 90), we get at once - 8J2 = P (a^4 - 4a,a, + 8a,«) + {aa^ - SAo, 4- Sco^ - <fao) («i^' - 803^' + 3a,ft' - a^fi') + (004 - 8*0, + SiJo, - <toi) (a'a,-85'aj + 3c'oi-rf'ao) For the biquadratic this gives the discriminant in the nsnal form^ Oq Ac, are then coefficients of the Hessian. Ex, 11. The expression ««l>'-4i«»t;Jf' + 6t«VJV-4«i;»3f +r*i) contains as a factor the square of the Jaoobian of u and v. If the division be effected, show that the quotient is 2 {P. J+ 6Zr (/)}. Ex. 12. The combinants P and Q of two cubics are each expressed in terms of the 8 and T invariants of the discriminant quartic, by a biquadratic whose quad- xinvariant vanishes. Ex. 18. If the roots of two cubics a, /3, y ; a', /S', y\ be connected by the relation •J{(a-a')(^-y')(y-/30} + »J{(^-^(y-a')(a-y')}4-V{(y-y')(a-/n03-a')} = O, •hew that « + Xv admits of being a perfect cube. Ex. 14. Determine the condition that a quartic may be such that its oovariant 12' with another quartic may vanish identically. ItiSi8fr=0. Ex. 15. The determinant form for J (Art. 202) is got by expressing that a quartic, whose covariant 12^ with « vanishes identically, has a double root. Ex. 16. If «. I- + 2a. ^ +...+ "o.-. ^ = «. h± + 2J, ^ +...+ rJ^. |-=«', concomitants of the system a^^ +...= «, b^ +...= », can be derived from their sources by repeating the operation ^ + d', and this operation on the coefficient of the highest power of x vanishes : as in the case of a single original function (Art. 147). Ex. 17. If we combine any quadratic with a cubic and form their Jacobian Art. 198, it becomes in the method of Art. 219c a plane, and as we vary (the quad- ratics) the chords of the twisted cubic the Jacobian varies, but always passes through a fixed point. This point is easily seen by the determinant form for the cubicovariant of a cubic (Art. 195) to be the point which corresponds to that cubicovariant. Ex. 18. In the system of cubic and. quartic the simplest linear covariant p. 219 vanishes identically if the cubic be of the form uvw and the quartic Au^ +J5v* + C%b/^, In general, the system may be reduced to the form Au^ + Bv^ + Cw^, -4'f**+jBV+ CV, where «« + 1; 4- w = 0, by the canonizant a', o, *, 3f» h'j bf Cj -ajy» c', c, dj xhi d\ <?, e, -»3 = UVV3 — 0. ( 227 ) LESSON XVIIL HIQHBB BINARY QUANTICS. 224. TTie Quintic. There are in all (including the quintic itself and four invariants) twenty-three forms. The invariants are /, K^ L of the orders 4, 8, 12 respectively, and a skew invariant / of the 18th. The discriminant R of the quintic Is not reckoned as a separate invariant, inasmuch as it is, as we shall presently see, a function (J* — 128-2") of the invariants J and K, Three of the covariants invite special attention, viz. the Hessian 12*, which if we take the quintic to be («)5jC,d,e,/3[a?,y)*) has for its value. * H^ (ac-J")aj*+ 3 {ad-'hc)x^y-\- 3 ((W+S(i-2c*)ajy +(a/+7Se-8c(Z)ajy +3 (J/+ce - 2d«) ajy + 3 {cf'-de)xj^^^{dif^ i)^ . There is a second covariant of the second order in the coefficients, viz. the covariant quadratic 12^, the 8 of the quartic emanant, which has for its value fif « (oe - 4&rf + Z&) a?" + (a/- 3Je + 2cd) xy + (J/- 4ce + W) y\ And thirdly, there is a covariant of the third order in the coefficients, viz. the canonizant, the determinant expression for which we gave Art. 169 ; that is to say, the covariant cubic which has for its factors the a;, y^ z of the canonical form. This covariant, which is also the T of the quartic emanant, has for its value [ace - acP-eb^+ ibcd- c') cc' + {acf- ode - jy + Jce + Jd" - cV) x^y ^'{adf-a^-hcf^-hde+(?e''cd')xy^'\-{bdf-U^'-'cy^^ 225. In studying the quintic we constantly use the canonical form aui^ '\'hy^-\-cz^ (where a?+y + « = 0), to which it has been 228 HIGHER BINARY QUANTICS. shown (Art. 168) that the general equation may be reduced. For this form the three covariants just considered are respectively 8 = hcyz + cazx + abxy^ T^cAcocyz. ' Differentiating the quintic with regard to x and y successively, we have u^ = ax^ — ca*, w, = hy^ - c«*. It is evident that the resultant of these two will be the discriminant of the quintic, and that the combinants of this system will be invariants of the quintic These invariants are then Immediately found from the expressions in Art. 223, where we must write for a and i, a and — c, for a and J', h and — c. We have (24), and therefore M= ; (13) = 1, (12) = - 1, (34) = - 1 , (14) = - 1, (23) = 1. We observe then at once that B vanishes. We can see, by counting constants, that any two cubics can be brought by linear transformation to be the two differentials of a single quartic ; but two quartics cannot be similarly brought to be the differentials of a single quintic, unless the condition jS = be fulfilled. Or It may be otherwise stated that this Is the condition that the quartics should be reducible to the form au* + iv* + cw*, aV + J V + c V. The comblnant A In like manner becomes JV + cV + a'i* - 2aJc (a + J + c). This, which we shall call e7. Is the simplest invariant of the quintic, and It may be obtained In other ways; viz. either by forming the discriminant of 8^ or the quadratic invariant 12* of H. In either way we obtain the general value of / = ay - IQahef-V 4.acdf-\- IGace" - Uad^e + UVdf -f 9JV - 12Jcy- l^hcde + 48Jd" + 48c'e - 320^". 226. The discriminant of the quintic may be obtained either from the theory of two quartics, or by direct elimination between the two differentials ax^ — c»*, hy*^ - c«*. When these THE QUINTIC. 229 vanish together, we may take aha as the common valae of ax*^ hy^^ cz*^ ; whence x = (6c)*, y = (ca)*, z = (aJ)*. Substituting in a; + y + isf = 0, we get the discriminant in the form (6c)* + [ca]^ + (a6)* = 0, or {&V+cV+a*J*-2a7>c(a+ 6 + c)}'-128a''JV(6c +ca + a6) = 0. Thus then we are led to the form for the discriminant /*— 128-K^ where K is the invariant of the eighth order in the coefficients, which for the canonical form is a^iV (Jc •{■ca-{' ah). This latter invariant may be otherwise defined : the invariant K is found by substituting in the usual way differential symbols for the variables, and operating with the square of the canonizant on the Hessian. This can easily be verified by the canonical form. Or else K can be found by forming the invariant /, as in Art. 198, of the co variant quadratic hcyz-\-cazx-\-ahQcy^ and the canonizant. In any of these ways the general value is found to be K= - 2aV - 2 jy '^ + a^pVe^ + 1 Xc^fhcde - hcfflce^ - hafVde + UaThd^ + 12ay Ve - 30ay icZ V - 30a/'JVe + ISa'^JcZe* + Ibl'cef^ - 21ay Vrf« - Z^ajc'de^ - UafFcd'' + 22a*cV -h 226V/» + l^c^fc^e + l^afUd - 48a'c(?c' - 486Vc^» - 27ayc?* - 27a/'c* + ISa^ V + 18JV/» + 133a/6Vcrf - 54a6'ce* - UVde^- 1 %afVd\ - 18a/JcV + 3aJWV + ^h^c^if" 220afbec^d^ + lOGaJcVc' 4- 106 J'c(7 V+ 93a/ Jcd?* + 93a/c*c?e - 30a6e'ccZ' - Z^Ve^df- ^ahed"" - 9 Jecy- 38acV - 386'tZy- 42a/c'd'.+ 8ac'c?V+ 86Vciy+ 6acV*6+ 66cVy + 27&V - %Wicd + 38JVc?'+ 38J'eV+25JVcW»- 576'ecJ* - 57 JeVrf+ 186^ '+ 18cV+ 74Jec'cZ'- 24JcW*-24c"c?'6+8cV*. The value of the discriminant in general can be derived hence^ or else, as I originally obtained it, from the formula (Art. 220) for the resultant of two quartics. We thus find 230 HIGHER BINABT QUANTICS. B^aT- 20a' f be - UOa'/'cd + 160 {ayce'' + a'/ Wrf) + 360 {a'fd'e + a^bd^ - 640 (a'/fe' + qfVc) + 256 (aV + J*/") - lOayjV - 1640ay"iec(i + 320 (a^/Jc'c + qfVed) - 1440ay* (M» + c'e) + 4080 (ayje'cT + aJ^bW) - 1920 (a"ie*i+ b*ecf^ + 2640ay*cV+4480 (a'/cW + afVcd^) - 2560 (a'cV + J*cf/") - 10080 (ayafe + afb(?d) + 5760 (a'c(?e' + b'c^) + 3456 (a*dy+ a/V) - 21 60 (a»£i*e» + JV/') - 180a/6V - 14920a/JVc(? + 7200 (oiVc + Ve^df) + OeOa/tJ'^ctf + ieV) - 600 (oJVcT + JVc*/ ) + 28480a/Jec'd«-l 6000(aieVrf+ J'eccT/) - 11520a/(Jcd* 4- c^de) + 7200 {Med' + JVc^) + 6400 {ac^e' + ^e?*/ ) + 5120a/c'd" - 3200 (aeVc?» + JVtP/) - 3375JV + 9000JVcrf - 4000 (JVcP + JVc") + 20006Vc'fr. The discriminant may also be expressed as Mows : Let ^ - ay* - 34a/Je + 76a/crf - 32ace* - 32 J"^^ - Uaed^ - 12 Jc"/ + 225iV - %20becd + 480 [bdP + c\) - 320c*d* ; B = 3ay * - 22a/J€ - Uafcd + 64 {ace^ + VdJ ) - 36 {aed^ + icy) - 45 JV + 2Qibecd ; C= aye + 2afbd - 9aJe' - 9a/b' + 32acrf€ - 1 8aef + 6JV -15&ec' + 10icc?, i? = ^a^df- 2aV - 9a/&c + abed + 18ac'6 - 12acc? + 6 jy -15yec+10y(f; and let C, iX be the fiinctions complemental to G and D (where all these functions vanish if three roots be equal), theu three times the discriminant is .15+ 64(7(7' -64i)2>'. 227. Quintics have also an invariant of the twelfth degree, which may be most simply defined as the discriminant of the canonizant. For the canonical form for which the canonizant is abcxyZj this discriminant is - a*6V. And, in general, this discriminant is — i, where the following is the value of 2/ as calculated by M. FaS. de Bruno. To save space in printing we THE QUINTIC. 231 • omit the complementary tenns. Thus {a*<?d^f*) stands for L = aV<f/* - 2 ia*cW/') + (aVe*/') - 6 {a*c^ef) + 16{a*ccl'ey*) - 14 {a*cde'/) + 4 (aW) + 4 (a*<f/') - 11 (aVey')4lO(oy V/) - 3 {a*d'e') + iaVcde'/' - 2 (a'J'cey ) + 6 (aV^ef) - 16 {a'b'dVf) + 14 (a'JW/) - 4 (a'JV) + 50a'6c'<?e/» - 82 [a'bc'dey*) + 32 (o'icV/) - 36 (a'6ci*/») + 30 [a'bcd'^/^ + 30 [a'bcd'ey) - 24 (a'&ofe") + 28 [a'bifef*) - 50 [a^b^e'/) + 22 (a'i<f e») + 16 (a'cV/) + na*c'6Pf + 50 {a'c'd'eY) - 16 (a'c'dey) - 16 (a'cV) - 54 {aVd'ef) - 46 (aV<f e'/ ) + 60 (aV<f e') 4 6 (a'ctiV) + 70 [a'cc^e*/) - 56 (a'cdfV) - 18 (aVV) + 14 (a'i^'e') + a'JV/* + 132a'6'cdey-50(a'iV/) + 14 {a'b'd^e'f) - 60 (aVd'eY) + 30 (a'J'<fe») - USa'bVcP^f + 48 (a'bVde*/) + 4 (a'6'cV) + 48 {a'Vcd^ef) + 2 {c^b*<xPe'/) - 6 (a'5'af e») - 62 (a'J't^y') + 90 {a'b'cPey) - 39 (o'6'<i*e*) - 112 ia'bcV/) - 82a'Jc'dV' + 170 {a'bc'd'e'f) + 104 (a'Jc'de*) + 108 {a'bc'(P/') + 42 {a'bc'd'eY) - 298(a'Jc'<f e*)- 242 (a'Jcd»e/) + 294 (a'WV) + 72 (a'J«P/) - 78 (a'Jd'e') + 164 {aVde'/) + 24 (a'cV) - 63aV<?/' - 394 (aV<f e'/) - 194 (a'cVe*) + 324 {aVcPef) + 440 (aV<?e') - 78 (aV<f/) - 428 (aVcT'e*) + 180 {d'cdPe) - 27 (a'O + ISab'e'f- 3Sab*cde*/+ 36 (a5*ce») + 204 {ab*cP^/) - 102 (oiVe*) - 308 {abVde') - 4SabVeP^f - 674 {ab'cd*ey) + 590 (oJ'af e*) + 128 (aJ'ti'e/) - 138 {ab'd^e') + 4 (aJ'cV) + 652 {cAVcPe*) + 714aJVd'ey + 498 (ahV^ef) - 1246 {abVd'e') - 224 [cffcd^f) + 516 (aJ'aTe') - 48 (aJ'd»«) - 136 [dbi^de*) - 1078a Jc*<i*«/- 206 (aJc*cP'e') + 342 {abc'tPf) + 804 {(d>c'dV) - 506 (aJcVe) + 90 (aJa?) - 16 {acV) + 220 {ac'cPe*) - 106ac'^/- 392 (ac°<^e')+222(ac*<f e) - iO(ac'cP) - 27bY + 234J''cde' - 32 (b'd'e') - llSbVdV + 246 {b*cdV) - 4 (J*d'e') + 866iV<?'e' - 550 (JVtiV) + 56 (J'ctTc) + 4 (J»i») - 139iV<?*c»+ 354 {bVd!'e) - 83 (iVd*) - 330Jc'(fe + 72 (JcV) - lec'cP. 232 HIGHER BINARY QUANTICS. On inspecting this invariant it will be seen that it vanishes if ft, c, d all vanish. Consequently the form ax^ + 5exy* +^*, to which Mr. Jerrard has shown that the quintic can be brought by a non-linear transformation, is one to which no quintic can be brought by linear substitution unless Z = 0. 228. We take «7, f , Z as the fundamental invariants of the quintic, and we proceed to show how all its other invariants can be expressed in terms of these. In the first place, it will be observed that the interchange in the canonical form either of x and y, or of x and z^ is a linear transformation whose modulus is — 1. Hence, if any invariant is such that when transformed it is multiplied by an even power of the modulus of transfor- mation, it must, for the canonical form, be unaltered by any interchange of a, ft, c ; that is to say, it must be a symmetric function of these quantities. If the invariant is multiplied by an odd power of the modulus, it must be such as to change sign when any two of the quantities a, ft, c are interchanged; it must therefore be of the form (a — ft) (ft — c) (c — a) multiplied by a symmetric function of a, ft, c. Now an invariant is in trans- formation multiplied by a power of the modulus equal to its weight. And (Art. 143) the weight of an invariant of the quintic, whose order is ti, is f tz. A quintic cannot have an invariant of odd order in the coefficients. If the order is a multiple of 4, the weight is an even number, and the sign of the invariant is unaltered by the interchange of x and y. If the order be not divisible by 4, the invariant is what we have called skew^ that is to say, such as to change sign when x and y are interchanged. Let us first examine the former kind, which we have seen must, for the canonical form, be symmetric functions of a, ft, c. Now, since e7=(ftc + ca4- aft)'— 4aftc(a + ft + c), ^ = a^hV [be ■{•ca-{-ab)j Z = a*ftV, (from which we infer jff = i{K'- JL) = a'ftV (a + ft + c),*) it follows that if we are ♦ The reader must be careful to observe that though, in the case of the canonical form, a^b^c^ {a + b-^ c), for example, is divisible by a^b^c^, we have no right to infer that in general H is divisible by Z, unless in cases where the quotient ahe {a + b-^e) has been also proved to be an invariant. FUNDAMENTAL INVARIANTS OF A QUINTIC. 233 given any quintic, and transform it to the canonical form by a substitution whose modulus is unity, the numerical values of the new a, b^ c are given by the cubic Now the order of any symmetrical function of a, J, c will be equal to its weight in the coefficients of this cubic, and when this weight is a multiple of 4, it is easy to see that the symmetric function is a rational function of J, f , L. Being given, therefore, any invariant whose order in the coefficients is a multiple of 4, it has been proved that we can write down a rational function of /, f , 2/, which, for the canonical form, shall have the same value as this invariant, and therefore be always identical with it. And since it would be manifestly absurd to suppose an integral function of the coefficients, to be equal to an irreducible fraction, it follows that every non-skew invariant is an integral function of J", JT, L. I£ we make the first three coefficients a, 5, e each equal 0, Jy Kj L all vanish. Hence when three roots of a quintic are all equal, these three invariants vanish.* If we make a, J, e^f all equal 0, e7 becomes — 32c'cZ", and i/, — 16c*dl*, and therefore J^—2048Z vanishes. Quintics therefore which have two pairs of equal roots must not only have the discriminant =0, but also /' = 2048i. 229. The simplest skew invariant is got by forming the resultant of the quintic ax^-\-hy^ -^-cz^^ and its canonizant ahcxyz. Substituting successively the three roots of the canonizant in the quintic, and multiplying together, we get for the resultant a*JV(J — c) (c-a) (a — J). This invariant, therefore, is of the eighteenth order. Previous to its discovery by M. Hermite,t * In general all the inyariants of a qnantic vanish, if more than \n of its roots be all equal. For when half the coefficients, counting from one end, simultaneously vanish, no term of the proper weight (Art. 143) can be made with the remaining coefficients. t See Cambridge and Dublin Mathematical Journal^ vol. ix. p. 172. M. Hermite works with a new canonical form, the x and y of which are the two factors of the quadratic covariant. The quintic then is supposed to be such that ae — ^bd+ Sc^, b/'-4ce + Sd^ both vanish, and the quadratic covariant reduces to xy. The advantage of this is that d^ the operating symbol thence derived is simply , ' , and some of the covariants HH 234 HIGHER BINARY QUANTICS. the possibility of the existence of skew inyariants bad not been recognised. I took the trouble to calculate this invariant, and the result is printed [Philosophical Transactional 1858, p. 455*), but as it consists of nearly nine hundred terms I cannot afford room for it here. The leading terms are cUd^f^ — cfcy ; in this, as in every skew invariant, the complementary terms having opposite signs, and the symmetrical terms vanishing. More- over if the alternate terms in any equation be wanting every skew invariant vanishes. For in this case the weight of each coefficient is even, but the weight of any skew invariant is an odd number. Thus / vanishes if 5, c?,y vanish ; that is to say, if the quintic can be reduced to the form x {a? — a*) (oj* — ^), in other words, if we consider the quintic as denoting five points on a right line, the vanishing of / is the condition that one of these points should be a self-conjugate point of the involution deter- mined by the other four. By the argument used, it is proved that every skew in- variant of a quintic must be the product of this invariant /by a rational function of J*, K^ L. - 230. The square of I being of the tldrty-sixth degree can be expressed rationally in terms of /, K^ L (Art. 228). The actual expression is easily found. By forming the discriminant of the cubic (Art. 228) we obtain the product of the squares of the differences of a, J, e in terms of e7, K^ X, and thus have PL = H'K' + ISHKr - 27Z* - IKT - 4^ ; or putting for H its value J {K^ - JX), and dividing by i, we have 16P = /JSr* + SLK' - 2J'LK' - 72JZX" - 432i' + J^L\ In the last equation of Art. 222 when we make jB = for two quartics derived from a quintic we find by the same article obtained by thus differentiating assume a very simple form. Notwithstanding, I have preferred using Sylvester's canonical form, which I find much more convenient. * Where the coefficient of b'd^ey shoviid have been printed 12500. COVARIANTS OF A QUINTIC. 235 E^D and by Art. 220 /+O=0; whence Jf" becomes the function written here* 231. We come now to the covariants. We have abready (Art. 224) mentioned the quadratic covariant 8 and the cubic covariant 71 Considering this system of a cubic and quadratic, we have (Art. 198) a series of covariants which give com- pletely all the covariants of the quintic which are not higher than the third order in the variables. The five invariants of Art. 198 reduce to four e7, K^ i, / already mentioned, the discriminant of the cubic, and the resultant of cubic and quadratic, both reducing to L. The four linear covariants of the system of cubic and quadratic give four linear covariants of the quintic, of the orders 5, 7, 11, 13, which for the canonical form are respectively ahc {bcx + cay + ahz) , abc {{bV+ a^hc) (y - «) + (cV+ Vac) («-«) + (a*6'+ c'oJ) [x - y)}, a'6V {5c (y - «) + ca (« - a?) + ah {x - y)}, a*5V {ax + by-^cz]. These are the only distinct linear covariants of the quintic. If we eliminate either between the first and last of these, or between the second and third, or between the first of them and the canonizant, we get Hermite's /; and if between the first linear covariant and the quintic itself we get I [J* — SK). Thus, then, if / vanish, or if e7* = BK^ the quintic is immedi- ately soluble, one of the roots being given by that linear covariant. Hermite has studied the quintic by transforming the equation, so as to take the first two linear covariants for X and y, when all the coefficients in the transformed equation are found to be invariants. The transformation becomes impos- sible wlien the two linear covariants are identical, which will be when their resultant JK+ dL vanishes. The system of cubic and quadratic have (Art. 198) three quadratic covariants, viz. in addition to ^8^ itself, the Hessian of r or a*5V (cc* + y* + «'), and the Jacobian of this and 8^ or a'JV {bcx (y — «) + cay («-«)+ abx {x - y)}. These are the only distinct quadratic covariants of the quintic. 236 HiaHEB BINARY QUANTIGS. Lastly, there are three cubic covarianta, vis. m addition to T itself, its cubic covariant a'iV (y — «)(« — oj) (« — y) ; and the Jacobian of B and T^ dbc \bcyz (y — «) -I- cazx (« — a?) + dbxy [x - y)]. These are the only cubic covariants of the quintic We have now enumerated fourteen forms, whose order in the Tariables is not higher than the third; adding to these the quintic and its Hessian, there are still seven forms to be mentioned. If we operate with 8 upon H^ we get a quartic of the fourth order in the coe£Scients, which only differs by a multiple of the square of 8 from abc (ooj* + iy* + ca*). A second quartic covariant is the Jacobian of this and 8^ or aJc{a"(J-.c)a;* + J«(c-a)y* + c"(a-J)«*}. These are the only two quartic covariants. We have a quintic covariant by taking the Jacobian of 8 and 17, viz. — aJc(y — «) {z — x) [x — y] [yz ■{■ zx •{• xy). A second quintic covariant is found by taking the Jacobian of U and the quadratic covariant a*6V («* + y* + «*). This gives a'JV [ax^ (y - «) + hy^ («-») + c«* [x - y)]. Of sextic forms there only is, in addition to the Hessian, the Jacobian of 8 and J9, or abc {oaf (y — «) H- iy^ {z^x)-\- cz^ {x - y)]. There is cue septic form, viz. the Jacobian of U and the simplest quartic covariant, or ahc { Jcy^' (y — «) + ca » V (« — a?) + ah a?y* [x-^y)]. And lastly, one nonic, uamely, the Jacobian of 27 and J7, or a'toy - aVx V + i"cy V - J'ay V + c W - c^hz^y"" + abcx*y*z^ {y — «)(« — x) [x — y). 232. The forms might also have been arranged, as Prof. Cayley has done, accordiog to their order in the coeffi- SOURCES OF COVARIANTS OP A QUINTIC. 237 cients. We give here, in his order, the leading terms of the less complicated. (1) u, Qaintic, a. (2) 8^ Quadratic, ae - 4M -f 3c\ (3) S; Sextic, ac - i*. (4) r. Cubic, ac6 - acZ" - J'e + 25ccZ - c'. (5) Quintic, ay- 5abe + 2acd + 8 JV - Qbc\ (6) Nome, aV-3aJc+25'. (7) Invariant J already given ; fourth degree in coe£Scients. (8) Quartic, a* (e" -d/)+a {Sbcf- Sbde - 4c'e + 4^?*) + 5b'ce + 2b'<P - 25y- 8JcV + 3c*. This differs by the square of 8 from the corresponding quartic covariant. Art. 231 ; and is 12^ of 2^ and u. . (9) Sextic, a* (^- de) - aby - 2abce + Aabtf - ac*d -f 3 J'e - 66'cd + Sbc\ (10) Linear, a* (c/* - 2de/+ 6*) + a (- jy - 4&ce/+ 8 JcZ*/ ) + a (- 2bde*-2c*d/+ 14cV) + a(- 22cd"e + 9d*)+ 6JV- 12i*C(^ - 15bW + lOJ'cf e + 66cy+ dObd'de - 20Jcd» - 15c*e + lOc'ef . (11) Cubic, a" {cef- Sdy+ 2de^) + a(- JV+ 14J(#- ll&ce') + a (-Jcf e-9cy+ 14cVe- 6ccf ) - 86"e^+ 95V+ 66*0^- 165*ccfe + 8i'i' + Zb(?e - 2JcV. This is the Jacobian of 8 and T. (12) Septic, a» (2cy- 5crfe + 3^1') + a (- 4JV+ 5i*^ + 5Jc*e) + a (- Ibcd^ + c'dl) + 26y- hVce - 26'cf + 85Vi-. 3&c\ (13) Quadratic, a* (- c^ + hcdef- 3ce' - 3^y+ 2cf e') + a (2JV* - 55W+ 36V - 5icV+ 7 Jcrf^ ) + a (- Jcrfe* - M'^e - c^df^r 6cV - 8cV''e + Zcd>) - jy " + 5JV+ 2JVy- 3i'cZe* - 86V(^- 4i'cV + iVcd^e - ft*rf* + 3 Jcy + 5&c»(?e - 4Jc*i' - %c\ + 2cV^ (14) Quartic, a' (- (^+ e^) + a* (3Jc/'+ 2jrfe/- 5Je»- 8cV) + a* (2cdy + 12c(ie»-6cfe)+a(-2&y'-2&*c6/-66''cf/fl3&''cfe") + a (20Jc'(^+ 4JcV - 52 J(kf e + 24id* - 9cy+ 20c'rfe - lOcV) + 6JV- Vib'^cdf- IhbW + lOJ'^cf e + 6&V/+ SOiVde - 205'af • 156c*e + lOJcV. 238 HIGHEB BINARY QUANTICS. (15) Linear, a' [cj^ - 4^« + 3ey) + a" (-J*/" - ibcef*) + a« (1 6Jrfy* + 4W6y- 15Je* - 6c'rf/" + 4cV/- 22ctZ V) + a' (26c(fe' + 9rfy- 12rf V) + a (7 J V" - 305*C(^' + VcJf) + a (- 74J>'dY+ 845*d6' + ISicy* + 160Jc*de/"- 98Jc"e') + a (- 20J<^y- 94J(^V + 5lM*e - 81cV+ 18c'rfy+ 140c'd6*) 4- a (- lOOcVe + 18cd») + 85*e^* - ISbV/^ eV'cy* + 325'cefe/* 4 45J'ce' + 112JVy"- 150J'*d;V - 65Ve/- 2845Vrfy+ SOJVde* + 3205W6 - 1205*rf' + 216JcV- 15JcV- 310Jc'<Z'e + ISOJc'd* - 54cy+ 90c"efe - 40c*rf'. (16) Quintic, a'(cd/"»-2cey+2rfV-*') + a'(-*"^'+2JV/) + a' (- 3Jcy * - 6 Jcefe/+ 13 Jce' - 8M»/+ 2 Jrf V) + a* (16cV- 2cVy- 38cV6» + 34ci'e - dd") + a (55 V" + 25'd6/- 12 JV - 2^bVe/+ 526Wy+ 75«cde') + a (- 225Ve - 52&c'd/*+ 34JoV + Sbc'd'e - Jcei* + ISc^) + a (- 25c*efe + lOc'rf') - 2 J^ + I0b*ce/- 2Sb*dy+ 306Ve* + Z^bVdf- 35ycV - 50&'c(?*e + 30^* - UbV/^ lObVde - 405Vd' - 1 560*^6 + I0bc'd\ (17) Invariant -ff* already given, 8*^ degree in coefficients. (18) Quadratic, 8*** degree in coefficients. (19) Cubic, 9*^ degree in coefficients. (20) Linear, 11*^ in coefficients. (21) Invariant L already given, 12**^ degree in coefficients. (22) Linear, 13**^ in coefficients. (23) Invariant /, 18^*^ in coefficients. For (18), (19), (20), (22) we refer to Prof. Cayley's Ninth Memoir on Quantics, Phil. Trans,j 1871, p. 17. 233. Prof. Cayley* has been led to consider in the theory of the quintic a new canonical form, which is obtained as follows : Taking for convenience the quintic to be (a, b, c, d, e, tjx, y)% * It has been already mentioned (p. 134) that the method of discussing coyariants by means of their leading terms or spvrcea was introduced by Prof. M. BobeitB See Quarterly Journal^ vol, iv. CELTEBIA OF REALITY OP BOOTS. 239 using small Roman letters for the coe£Scients, suppose in the first instance that a, i, c, d^ 6, f denote the leading coefficients of the first six covariants of Art. 232 respectively, thus o = a, <j = ac-b«, e = a«f-6abe + 2acd+8b2a-6lxi», * = ae - 4bd + 8c«, <?= ace-ad*-b«e-c»+2bcd, /=:aM - 3abc + 2b», where — y = a' (ad— Jc)+4c' identically, so that any rational and integral function containing f can always be expressed as a function linear in regard to f. This being so, we have the function i (a, b, c, d, e, f )(x - by, ay)» = «» + 10a^y« (ac - b«) + 10a:«y» (aM - 8abc + 2b«) + hx^ (a»e - 4an)d + Bab^c - 3b*) + y» (a*f - Sanw + 10an)2d - lOaVc + 4t>»), and forming the values of c^h — 3c' and a'e - 2o/*, this is found to be = (1,0, c,/, a*5 - 3c», dfe - 2c/3[ar, y)". The last-mentioned function, considering therein a, i, c^e^f z& denoting not the leading coefficients, but the covariants them- selves, and (a;, y) as variables distinct from those of the quintic and its covariants, is the canonical form in question. Using in like manner d to stand for the covariant, we have between the covariants a, 5, c, d^ 6,y the foregoing identical equation -/* = a'(arf-5c) + 4c", which Is to be used to reduce functions of the covariants so as to render them linear in regard to^. 234. Criteria for the reality of roots of quintics. It ought to have been stated earlier that the sign of the discriminant of any quantic enables us at once to determine whether it has an even or odd number of pairs of imaginary roots. Imagine the quantic resolved into its real quadratic factors, then (Art. 110) the discriminant of the quantic is equal to the product of the discriminants of all the quadratics, multiplied by the square of the product of the resultants of every pair of factors. These resultants are all real, and their squares positive ; therefore, in con^dering the sign of the discriminant, we need only attend to the discriminants of the quadratic factors. But the square of the difference of the roots of a quadratic is positive when the roots are real, and negative when they are 240 HIGHER BINABT QUANTICS. imaginary. It follows then that the product of the sqaaiies of the differences of the roots of any quantic is positive when it has an even number of pairs of imaginary roots, and negative when it has an odd number. We have been accustomed to write the discriminant giving the positive sign to the term which is a^power of the product of the two extreme coefficients. This will have the same sign as the product of the squares of differences of the roots when the order of the quantic is of the form 4m or 49n + 1, and the opposite sign when the order ifi of the form 47n -f 2 or 4m + 3. We see then, in the case of the quintic, that if the discriminant be positive, there will be either four imaginary roots or none ; and if the discriminant be negative, there will be two imaginary roots. It remains then further to distinguish the cases when all the roots are real, and where only one is so'. 235. In order to discriminate between these remaining cases, there are various ways in which we may proceed. The fol- lowing* are, in their simplest forms, the criteria furnished by Sturm's theorem. Let cT^be the quartic invariant as before, and jBf =J'-ac, 8 = ae"4:hd+Sc*j T=:ace'^2bcd-ad^-eb^-c\ M= aV - a*d/+ Zahcf- Sabde + 4acrf* - 4ac'6 - 2 jy 4- 5Vce + 2b^d* - Shc^d + 3c*, then the leading terms in the Sturmian functions are proportional to a, a, H, 5J?S+9ar, -S7+ 125ilf+4/8'-216r*, the last of course being the discriminant ; and the conditions fiimished by Sturm's theorem to discriminate the cases of four and oo imaginary roots, are that when all the roots are real the three quantities jff, 5HS+daTj — S/+&C. must all be positive. ♦ These values are given by Mr. M. Roberts, Qiuxrterly Journal^ vol. iv. p. 175. The reader who may use Prof. Cayley's tables of SturmlMi functions {Philosopliical Transactions^ vol. OXLVII. p. 735) must be cautioned that the fourth and fifth functions are there given with wrong signs. M is already written as (8) in Art. 232, and is connected with D in Art. 226 by the equation D = S^ — 3M. In fact the expression for the discriminant R thett given is 9R = 25 J^ - 192 {B^D^ - AD^D^ + Si)^*), where the covariant whose sooioe is i) is written D^Jt4D^a?y+&c., and i>=A» C!= 2^i, 3B + A = - 107, ^B-A -^^9 C-2B^,B' = B^, STUEM'S functions fob a QUlNtlC. 241 236. We maj apply these conditions to the canonical form (c - a) aj* -f 5ca;*y + lOcxy + lOcajy + 5cajy* + (c - J) y*, in which case the equality of all bat two of the coefficients renders the direct calculation also easy. We easily find then that the constants are c — a^ c—a^ acj — aV; and the fourth being essentially negative, we need not proceed further, and we learn that the equation just written has always imaginary roots. We find then that when the invariant 2/ of a quintic is positive, the roots of the equation cannot be all real. For L being, with sign changed, the discriminant of the canonizant, when L is positive, the roots of the canonizant are all real, and the quintic can be brought to the canonical form by a real transformation. When L is negative, two factors of the canonizant are imaginary, and the canonical form is a(-2aj)'+ {c- J V(- 1)} {^ + y V(- !)}• + {c+cZV(-i)}{«-yV(-i)r, which, expanded, is dt^ + 5cy^x - lOd/x^ - lOcy V + &dyx* + (c - 1 6a) x\ Writing for brevity c* + rf* = r*, I find for this form the Sturmian constants to be rf, dj r", /, r'(— 4a*d* + 20acr" + 5/), and it would seem that the discriminant being positive, the roots are aQ real if d and — 4a'rf' + 20acr' + 5/ are both positive.* 237. In practice the criteriaf furnished by Sturm's theorem are more convenient than any other, because the functions to be calculated are of lower order in the coefficients. It is, how- ever, theoretically desirable to express these criteria in terms of the invariants, and this is what has been effected by different * I give this reBult, though suspecting its accuracy, because it seems to me to disagree with the theory derived from the other methods. t It may be noticed that there is no difficulty in writing down a multitude of criteria which might indicate the existence of imaginary roots; for any symmetric function of squares of differences of roots £ (a — /3)^, &c. must be positiTe if aU the roots are real. We can without difficulty write down such functions which are also inyariants ; and which, if negative, show that the equation has imaginary roots. But then these may also be positive when the roots are imaginary, and the problem 28 to find some criterion or system of criteria, some one of which must fail to b6 Batisfied when the roots are not all real II .242 H16HEB BINARY QUANTICB. methods by Hermite and by Sylvester. We proceed briefly to explain the principles of Sylvester's method, which is highly ingenious. We have seen already that when the invariants J^ f , Zt are given, then a, hj c of the canonical form may be determined by a cubic eqtiation ; and we can infer that to every given system of values of /, K^ L will correspond some quintic. But to every system of values of «7, Kj L will not correspond a real quintic. In fact, we have seen, Art 230, that the /, K^ L^ of every quintic with real coefficients, are sucb that the quantity O is essentially positive, where O is JK"^ + ^LK' - 2rLK^ - 12JUK'- 432i» + rL\ For O has been shown to be the perfect square of a real function of the coefficients of the general quintic, viz. aVy*+&c., this being the eliminant of the quintic and its canonizant, and therefore necessarily real. We may in the above substitute for JTits value in the discriminant from the equation J^— 128£'= Dj and so write (7, Ji)* - 4 (/' + 2«i) LP + (6«7» - 29-2"i) J^LP - 4 (/*- 61.2^*2; - 9,2*»i*) JD + (e7^- 2"i>» (/»- 27.2"X). If now, to assist our conceptions, we take /, -D, L for the coordinates^ of a point in space, then (7=0 represents a surface; and points on one side of it, making O positive, answer to real quintics, while points on the other side^ making O negativef, answer to quintics with imaginary coefficients. 238. Now, in the next place, we say that if the coefficients in an equation be made to vary continuously, the passage from real to imaginary roots must take place through equal roots. For, let any quantic <f>[x) become by a small change of coefficients ^ {x) + e^ [x)j where & is infinitesimal, and let a be a real root of the first, a + ^ a root of the second ; then we ^ ■ - -■■■--■- _ — _^ * Sylvester takes L in the nsaal direction, of a;, JFef ^ and JD of z^ t Points for which G = answer to real quintics^ and it is easy to see that in this case the equation admits of linear transformation to the recarring fozm. Far we have proved that when (? = two of the coefficients of the canonical foxm aie eqoiL The equation is therefore of the form ax^ + caf' + b {x + yY — 0^ METHOD OF SYLVESTER. 243 bave 0(a + /i) + e^(a) = O; whence, since ^(a) = 0, we have A^ (a) + e^ (a) = 0, which gives a real valae for A, The con- Becutive root a + his therefore also real. But if if/ (a) vanishes as well as ^(a), the lowest term in the expansion of <]}{a + k) will be A*, and the value of A may possibly be imaginary. When, therefore, the original quantic has equal roots, the cor- responding roots of the consecutive quantic may be imaginary. It follows then, that if we represent systems of values of -Jj J9, L by points in space, in the manner indicated in the last article, two points will correspond to quintics having the same number of real roots, provided that we can pass from one to the otherivithout crossing either the plane D or the surface O^ If points lie on opposite sides of the plane J9, we evidently cannot pass troin. one to the other without having, at an inter- vening point, i> = 0, at which point a change in the character of the roots might take place. If two points, both fulfilling the condition G positive, be separated by sheets of the surface Oj we can not pass continuously from one of the corresponding quintics to the other; because when on crossing the surface we have O negative, the corresponding quintic has imaginary coefficients. But when two points are not separated in one of these ways, we can pass continuously from one to the other, without the occurrence of any change in the character of the corresponding quintics. Now Sylvester's method consists in shewing, by a dis- cussion of the surface (7, that all points fulfilling the condition G positive, which he calh facultative points, may be distributed in three blocks separated from each other either by the plane D or the surface G, And since there may evidently be quintics of three kinds, viz. having four, two, or no imaginary roots, the points in the three blocks must correspond respectively to these three classes. I have not space for the elaborate investigation of the surface O^ by which Sylvester establishes this; but the following is sufficient to enable the reader to convince himself of the truth of his conclusions. 239. One of the three blocks we may dispose of at once, viz. points on the negative side of the plane D^ which we have 244 HIOHEB ItlNAST QUAKTICS. seen (Art. 234) correspond to qointics having two imaginuy roots. Next with regard to points for which D is poaitiTe. We have Been, in the last article, that a change in the character of the roots takes place only when i) = ; our attention is therefore directed to the section of Q by the plane D. We see at once, by making jD = in the value of 6 [Art. 237), that the remainder has a square factor, and consequently that the surface Q touches D along the curve J^'-2"i, and cuts it along t/'— 27 .2"i. Now, if a surface merely cut a plane, the line of sectioQ is no line of separation between points on the same side of the surface. If, for example, we put a cup on » table, there is free communication between all the points inside the cup and between all tboee outside it. But if a plane touch a surface, as, for instance, if we place a cylinder on a table, then while there is still free communication between the pointe inside the cylinder, the line of contact acts as a boundary line, cutting off communication as far as it extends, between points outside the cylinder on each side of the boundary. Now Sylvester's assertion is, that if we take the negative quadrant, viz. that for which both J and L are negative, and if we draw in the plane of xy the curve J* — 2"L, then all facultative points in that quadrant, lying above the space io- clnded between the curve and the axis i = 0, form a block completely separated from the rest, and correspond to the case of five real roots. 240. In order to see the character of the surface, I form the discriminant of G considered as a function of A'^, which I find to be —V (/^+27iv)'. Consequently, when both J and X are negative, the diacrimiuaut is negative, and the equation in K has only two real roots, To every system of values, therefore, of J and i correspond two values of A", and consequently two values of D and the surface is one of two sheets. Now I say that it is the space between these sheets for which G is positive. In fact, since G is JD* + &c., it may be resolved into its factors J {D - a) (B - ^) {{D - yf + S'] ■ and since J is supposed to bo negative in the space under consideration, D must evidently be intermediate between « and ;3 in order that G should be positive. DISCUSSION OP METHOD OP SYLVESTER. 245 Now the last term of the equation being (c7'-2''2i)* (er-27.2*°Z}^ if /' be nearly equal to 2"i, will be of opposite sign to i, or in the present case will be positive. And the coefficient of Tf" being negative, we see that on both sides of the line J* = 2"2y the values of D are, one positive and the other negative, that is to say, the two sheets of the surface are one above and the other below the plane D. But I say it is the upper sheet which touches B along J^ — 2"2y, This may be seen immediately by looking at the sign of the penultimate term in the equation for i), by which we see that when the last term vanishes, the tWo i?oots are and negative. The theory then already ex- plained shows that the curve J^ = 2"Z acts as $t boundary line cutting off communication in that direction between facultative pbints on the upper side, of D. But, again, communication in the other direction is cut off by the plane ii = 0. For when we make L positive, the discriminant becomes positive, and the equation in D has either four real or four ima^nary roots. But the first Sturmian constant is proportional to L (J* -f 12L\ which, when J is negative, and L positive and small, is negative. Immediately beyond the plane L^ therefore, the equation to determine D has four imaginary roots, or the surface does not exist. The facultative points, therefore, lying as they do mthin the surface or between its sheets, are cut off by the plane L^ on which the sheets unite, from communication with points beyond it. Thus the isolation of the block under consideration has been proved. I need not enter into equal detail to prove that. all other facultative points have free communication inter se. The line pf contact 2"2i — e7' is no line of separation in the quadrant where J and L are both positive. For then it is seen, as before, that it is the points outside the two sheets which are facultative, and not the points between the surface and touching plane. The result of this investigation is, that in order to have all the roots real, we must have the quantity 2"i/ — /' positive,* * Sylvester has inadyertentlj stated his cooidition to be that 2"X — /* is negative. It is easy to see, however, that what he has proved is, that this qmuitity must be positive. For the block which he has described lies on the side of the curve 2"Z/-*J» next to the axis L =• 0. But when £» is and J negative, 2**.L-/» is positivB. 246 HIGHEE BINABY QUANTICS. and L negative, which also infers J negative. If either con- dition fails, our roots are imaginary. It is supposed that in both cases D is positive. 241. We have seen that the cylinder parallel to the a^ of z and standing on the curve 2^2/ — J^ does not meet G above the plane D ; the two values of z being one 0, the other nega- tive. Any other surface then standing on the same curve and not meeting O would serve equally well as a wall of separation between the two classes of facultative points. For, all the points between the cylinder and this surface would be non- facultative, and therefore irrelevant to the queistion. Sylvester has thus seen that we may substitute for the criterion 2^^ii— c/', 2"i — e7' + /i«/2), provided that the second represent a surface not meeting O above the plane D. And on investigating within what limits /a must be taken, in order to fulfil this condition, he finds that /a may be any number between 1 and — 2. He avails himself of this to give criteria expressed as sym- metrical functions of the roots. In the first place S(a-i8r(/3-7)'(7-ar(8-er is an invariant (Art. 136), and being of the same order and weight as J can only differ from it by a numerical factor, which factor must be negative, since this function is essentiallj positive ; and / we have seen is essentially negative when the roots are all real. And secondly, the symmetric function 2 [a- fir (-8-7)' (y-a)' (s "«)* (6-/3)* (e-7)*(8-a)*(S- /S)* (8-7)', (the relation of which to the other may be seen by writing it in the form D"^ (a - /Sp (/3 - 7)"" (7 - «)"* (« - e)"*, where D is the discriminant], is also an invariant, and of the twelfth order. It must therefore be of the form aJ^ -{- fiJB ^ yL. Now, by using the quintic* x (a?* — a^) (a* - J*), the symmetric function * It was observed, Art. 229, that the characteristic of this form is that Hennite'a inyariant / vanishes, hence it may be safely used in calculating any invariant fonctioa whote order is divisible by 4 and is below 36, since such forms cannot contain /, but though this form may be safely used in this case, it cannot always be safely Q9ei For when a linear factor of a quintic is also a factor in the sextic covariant of ^ remaining quartic, a relation must exist between the invariants. INVARIANTS OF PRODUCT OF QUARTIC BY A FACTOR. 247 may easily be calculated and identified with the invariants ; and the result is that its value is proportional to 2"ii — J' + ^JD. Since then the numerical multipUer of JD is within the pre- scribed limits, it may be used as a criterion, and Prof. Sylvester's result is, that the two symmetrical functions mentioned are such that not only are both positive, as is evident, if the roots are all real, but also if both are positive, and D positive, the roots miLSt be all real It ought to be possible to verify this directly by examining the form of these functions in the case of an equation with four imaginary roots. 242. I have also tried to verify these results by examining the invariants of the product of a linear factor and a quartic, [ax -{- ^y) [x^ •\-^ma?y* -{-y^)'^ these being necessarily covariants of the quartic (Art. 201). The coefficients of the quintic are then 5a, /3, 37na, 3mi8, a, 5/3 ; and I find for the J of the quintic^ 48 (8)Sff- 3 TU)^ or 48 times (5m + 27m'J (a* + ^) + (8 - 18m* - 54m*) a'/S". Now the roots of the quartic are all real when m is negative, and when ^rn? is greater than 1. On inspection of the value given for /, we see that when m is negative every term but one is negative. Giving then m its smallest negative value — ^, J is negative, viz., - 144 (a'* — ^8')* ; and J\%d fortiori negative for every greater negative value of m. Or we may see the same thing by supposing ^9 = 0, when we have only to lo(^ at the coefficient of the highest power of a in 8/Sff— ZTU^ which is - 8 (J* — ac) 8 — ZTa, But now if we call the three Sturmian constants A^ B^ (7, viz. the value given for J becomes —QAS-^Bj which is essentially negative when the roots are all real. The invariant X, according to my calculation, is 54 [SSH^ 3 TUf - 6400 {8^ - 27 T') {4JI'' -8HU^+ TU') + 150 (5'-27r«) U'{S8E+ 15rZ7) -4050Z7»;8'(2fiff-3rZ7), whence 2"ii — /' differs only by a positive constant multiplier from -128()S'-27r«) [4:H'^S8HU'+TD') + 3 ()S*- 27 2'») U' {SSn+ UTU) - 81 U'8' (2fifl- dTU], 248 HiaHB£ ttNABT QUAMTICB. Wiitmg 0^cfd'-Sabc + 2b% the coeffident of the lughest power of a in this is 128 Ce* + S\a*S^ + 45a« CB - 54a" 08A. All the terms of this but one are positive when the roots are all real, bnt as there is one negative term, is it not obvioiis^ on the face of the formula, that the whole will be positive when the roots are all real. Still less that if this formida be positive and J negative, the roots are necessarily all real. Therefore, although no doubt Prof. Sylvester's rule may be tested by the process here indicated, to do so requires a closer examination of this formula than I am able to give.* 243. Prof. Cayley in his Eighth Memoir on Quantics {Phil. Trans. 1867), proceeded by a method a little different from that described above. Adopting as coordinates » = — j8 — > y 7^1 ^^"^i then from the foregoing equation ler = JK^ + SLK^ - 2J'LK* - 72JI«-Br- 432i' + rPL^ where, K= li^ (t/* — J9), we deduce without much difficulty p 2.2" -J « - 3iB' - a?" + y (72aj" + 205aj + 125) + / (- 29aj - 17) + /(-»- 9) + y*-2 = ^ {xj y) suppose ; or, since « = «/", we have z^ (ar, y) = 2 . 2"* jg = positive, and the surface G^ = may be replaced by z^ (a;, y) = ; that is, by the plane « = and the cylinder ff> (a?, y) = 0. The configura- tion of the regions into which space is divided by this surface depends only on the form of the curve ^ (a;, y) = (Prof. Sylvester's " Bicorn "), which is the section of the cylinder by the plane 2; = 0, and the discussion as to the reality of the roots may be then effected by means of the plane curve alone ; the results, of course, agree with those obtained above. ♦ The verification, however, is easy in the particular case x{a:^ + ^mxh/^ + y*). We have then /= 48m (5 + 27w2), L = 12m (5 - 9m^y j 2"L - J* proportional to m (1 - 9m2) (50 + 46m^ + 648w* + 729m«). Thus, when m is negative, and 9»i« > 1, we have / and L negative and 2"L — J^ positive. The latter is positive for imaginaiy roots only when m is positive, but in this case J is positive. The imaginary roots must, therefore, be detected by one criterion or other. hermite's typical form. 249 244. It has been already mentioned (Art. 231), that M. Hermite has made use of the fact, that the quintic as well as every equation of odd degree is reducible to a forme-type^ in which the x and y are linear covariants and the coefficients are invariants. It follows immediately, that by applying Sturm's theorem to the forme-type^ the conditions for reality of roots may be expressed by invariants. Hermite extends his theorem to equations of even degree above the fourth, by the method indicated in Art. 248. Writing *P -SK=^My JK-\rdL==N] and Q a numerical multiple of Hermite's 7, such that Q" = JK'JiP - 2MNK[J^ + 12K) +JN'{J^ + 12K) - 48J^, then the coefficients o( the forme^type are D = J^KW - JMN[J^ + SOiT) + N' (42e7^' + 1442r), E=Q[J'M^2UN), F=:J'KAr-'J'MN{J^+^2K) + NV{5iJ''^2SSK)^U52N\ Thus the first Sturmian constant B* — ACi8 found to be The Sturmian constants being essentially unsymmetrical, there seems no reason to expect that the discussion of these forms would lead to any results of practical interest. The coefficients o{ the forme-typej as M. Hermite remarked, satisfy the relations AJ'"2CJ+E=0, 5e7»-2Z>e7^+^=-1152JV^, -4JE-4J5i> + 3(7» = -12*Ar», AF-SBE^2GD = 0, BF- 4.CE+ 3Z?« = 12*e7Ar*. Thus then the quadratic covariant is N^{x*-Jy*)] and operating with this on the quintic, we get the canonizant in the form N' [AJ^ C, BJ- 2), OJ- E, DJ^ FJx, yY ; the coefficients inside the parentheses being all further divisible by N, Hence we have A CE-^ 2BGD - AD" - EB' - C'* = - 4 . \2'N'Q, 250 HIGHER BINARY QUANT1C9. and the second Sturmian constant Is got immediately hj substituting in the formula of Art. 235, the values just found for B^'-AG, AE--ABD + 3C\ ACE+2BGD-'&c.* 245. The Tschimhausen transformation consists in taking ^ new variable then there are n values of ^ corresponding to the n values of a;, and the coefficients ot the new equation in y are readily found in terms of those of the given equation by the method of symmetric functions, the first for example being 05^+ /Ss^+ys^+Scc The coefficient of y"~* is evidently a linear homogeneous function of a, iS, &c., that of y*~* a quadratic, of y*~^ a cubic function, and so on. In the case of the quintic, the transformation is y = a + iSar + yx^ + Sx', and we have four constants a, ^8, 7, S at our disposal. Mr. Jerrard pointed out that the coefficient of y' being a quadratic function of a, /3y 7, 8 was (Art. 165) capable of being written as the algebraic sum of four squares^ say i* - w* + v" - w\ It can therefore be made to vanish, by assuming two linear relations between a, ^8, 7, S; ^ — w = 0, v — w^O, If we combine with these two that linear relation which makes the coefficient of y vanish, we have three relations enabling us to express three of the constants a, ^8, 7, S linearly in terms of the fourth. We can then by solving a cubic make the coefficient of y^ also vanish, or else by solving a biquadratic make the coefficient of y vanish. In this way Mr. Jerrard showed, that by the solution of equations of inferior orders, a quintic may be reduced to either of the trinomial forms y + J^ = c, or y^ -f hy^ = c. The actual performance of the * The coeflScients of the forme-ti/pe of the qomtic were given by M. Hermite {Cambridge and Dublin Mathematical Journal^ 1854, vol. IX. p. 193)^ and re-calenlated by me before I found out the key for the translation of Hermite's notation into Sylvester's, which is A = /, J^ — — Ky J^ — JK + 9X. The discussion of the invariantive characteristics of the reality of the roots of a quintic was originally commenced by M. Hermite in the same classical paper, and was resumed by him in his valuable memoir presented to the French Academy, t. 62, 1866. His result, in our notation, is that the roots are all real, when the dis- criminant being positive, we have also positive Z, 2^^L-J^+JDf and K{JlA-K^iSLK It seems to me that this result is superseded by the greater simplicity of Prot Sylvester's criteria. tschirnhausen's transformation. 25i transformations would be a work of great labour, but M. Hermite showed how, by somewhat altering the form of substitution, we can avail ourselves of the help of invariants. If we have to transform the equation oa:* + Jx""* + cx*^ + &c., Hermite assumes y = a\ + [ax + J) a + [aa?-{' &« + c) ;8 + (aa5'+ hx^-\- co; + e?) 7 + &c , then in the first place the transformed equation will be divisible by a ; and secondly, if the given equation be linearly transformed, and if the corresponding substitution for the transformed equation be Y^Ax: + [AX + B)a! + [AX^ + BX-\- (7)/3' + &c., then he has shewn that the expressions for a\ ^, &c in terms of a, iS, &c. involve only the coefficients of linear transformation, and not those of the given equation. It is not so with respect to the first coefficient \, which we have therefore designated by a special letter. But the theory of linear substitutions will be directly applicable to all functions of the coefficients of the transformed equation which do not contain \. Such, for example, will be all symmetric functions of the differences of the roots of the new equqition, since, on subtracting y^ = a\ + {ax^ + J) a + &c., y, = a\ + [ax^ + J) a + &c., X disappears. Or, what comes to the same thing, if we take X such that the coefficient of t/"'^ in the new equation shall vanish, then the theory of linear substitutions is applicable to all the coefficients of the transformed. I give Cayley's proof of Hermite's theorem, and, after his example, take, to fix the ideas, the quartic (a, J, c, rf, ejx^ l)\ Then, as we have used binomial coefficients, we shall write the equation of transformation y = a\ + (oa; + 4 J) 7 -- (ax^'\- 4 Ja? + 6c) i8 + (aaj'+ 4 JajV Gca? + 4rf) a. Adding the 4 values of y, and employing Newton's formulae ibr the sums of powers of the roots, we see that the coefficient of y""* in the transformed equation will vanish if a\ + 3 J7 - 3c/3 + rfa = 0. e5S HieHEB BINARY QUANTIGB. This reduces the value of ^ to (ax + b) y - {ax* + Abx-i-Sc) ^ + (oa* + 4 Ja?" + 6cic + 3rf) a. « In general It will be obserred, that in this substitution all the terms have the binomial coefficients corresponding to the order of the given equation, except the terms not involving x^ which have the binomial coefficients answering to the order one lower. 246. Now what is asserted is, that all the coefficients of the transformed equation will be invariants of the system r (a, i, c, rf, ejx, y)\ (a, /3, yjx, y)\ and of course if we regard y as constant, the whole transformed function will be such an invariant. This will be proved bj shewing that it is made to vanish bj either of the operations Ml d ^ d ^ ^ d d f^rkd d\ Let the general substitution be y = F, and let F^, F^, &c. be what F becomes when we substitute for x each of the roots of the given equation, the transformed in y is the product of the factors y — Fj, y — T^, &c., and it is sufficient to prove that each of these factors is reduced to zero by this differentiation. We may, as in Art. 64, write the first part of the first operation ^, and in order to calculate -^ , we must find -^. Operating on the given equation, we get (a, i, c, dJx^lY-^-^ (a, J, 0, ^x, 1)^ = 0, or ^ = - 1. The part then of the differential of V which depends on the variation of a; is - [ay - [2ax + 4 J) /3 + (3aa?* -f 86a; + 6c) a}, and the part got by directly operating on the a, i, &c. which explicitly appear in F is 07 - (4aa; -f 66) ^ +. [^ax^ + 126a; + 9c) a. INVARIANTS IN QUADRATIC TRANSFORMATION. 258 Adding, we have dV ■ , Tx zi / 1 .7 «v r dV ^r,dV\ -^ =-2(aaj + J)i8+(aa;" + 4&a: + 3c)a = -fa^+2/3^J, which proves that the effect of the first operation on V is zero. In like manner, for the second operation, we have, by performing on the original equation, dx (a, J, c, d$Xj 1)' ^ + a? (J, c, dj ejxj 1)" = 0. But the original equation may be written X (a, J, c, djx^ ly + (J, c, rf, e3[a?, 1 )' = 0. dx dV Hence -^ = a?'. The part of ^- due to the variation of a; is drj '^ dfi therefore aaj'7 - (2aic' + 4 Ja:') /S + (Soar* + 8 Ja?' + 6ca;') a. The remaining part is (4Jaj + 3c) 7 - (4Jaj* + 12ca; + 6c?) ^8 + (4Ja;' + 12ca;' 4 l^dx + 3e) a. Adding, the coefficient of a vanishes in virtue of the original equation, and the remaining part is found to be '^K^d^'-'^d^)^ which completes the proof of the theorem. 247. When this transformation is applied to a cubic, if we consider a, fi as variables, the coefiicients of the transformed equation in y will be covariants of the given equation. The transformed in fact has been calculated by Prof. Cayley, and found to be tf^SHy+J^ where -ff is the Hessian {ac—F) a*+ &c., and J is the covariant (Art. 142), (a*e?- 3aJc + 2 J") a' + &c. Prof. Cayley has also calculated the result of transformation as applied to a quartic. Take the two quantics, as in Art. 212, (a, J, c, d, ejar, tf)% (a, ^8, yjx, y)\ and let C denote the skew invariant of the same article, p. 203 ; let 8 and T denote the two invariants of the quartic; also let 2' = 6 A, then the transformed function in ^ is 2^* + 6 (S + S^) y'-\'^Gy'^ 8(l>^ - 32" - 6SSA + ISTi^A. 254 HIGHEB BINABT QUANTICS. Prof. Caylej has also calculated the 8 and T of the trans- formed equation. In making the calculation^ it is useful to observe that since the square of Jj from which G was derived (p. 204), can be expressed in terms of the other invariants, so also may the square of G] the actual expression derived from his being in our notation - (7= r^'- ;8S^'- 9 (22 + 8A) TA<^+(2S + 35A)'S + 54T*A'. The result then is that the new 8 is 5^* + 3/S'A*+18rA^, and the new T is T<f>' + 8*A<I>^ + d£i'8T<l> + A' (54T« - 8'). Finallj, he has observed that these are the 8 and T of U<l) + GZTA, as may be verified by the formulae of Art. 210. It follows, then, that the effect of the Tschirnhausen transforma- tion is always to change a quartic into an equation having the same invariants as one of the form U+\Hy and, therefore, reducible by linear transformation to the latter form. The foregoing results in a different notation are reproduced, and the corresponding results for the quintic are obtained in Prof. Cayley's Memoir on Tschimhausen's Transformation, Phil. Trans.^ vol. CLII. (1862). 248. The following is the form in which M. Hermite presented his theory, and applied it to the case of the quintic. Let w be a quantic (a;, y)" ; w^, u^ its differentials with regard to X and y ; let <^ be a covariant, which we take of the degree w — 2 in order that the equation we are about to use may be homogeneous in x and y ; then the coefficients of the transformed equation, obtained by putting « = —, are all invariants of ti. The equation in z is got by eliminating x and y between zu^-'t/<f> = Oy and w = 0, or, what comes to the same thing, zu^ + x<f> = Oy which follows from the other two. If we linearly transform x and y, the new equation in z is got, in like manner, by eliminating between « Z/^ — Z4> = 0, « Z7, + X4> = 0. But, if x = \X-\- fiY^ y=^\'X+f/Y^ A = X/u,' — \'/A, we have AX = fi^x — fiy^ A Y= \y — X'o;, and Art. 130, U^ = \u^ + \'w,, Z7, = fiu^ + /^'t*,, and, since ^ is a covariant, we have 4> = A^<f>. Making these substitutions, the equation in z^ corresponding to the transformed equation, is got hekmite's theory. 255 by eliminating between z {\u^ + \\) - A*-*^ (\y - X'x) = 0, z (jjLu^ + fi\) + A*"'^ [fi'x - fiy) = 0. Multiply the first by fi\ the second by \', and subtract, and we have Azu^" A*y^==0. In like manner, multiplying the first by fly the second by X, and subtracting, we get Azu^ + A*x^ = 0. In other words, we have the two onginal equations, except that z 18 divided by A*"\ Consequently, the equations in z con^e- sponding to the original equation, and to the same linearly transformed, only differ in having the powers of . multiplied by different powers of the modulus of transformation A, and therefore the several coefficients of the powers of z are invariants. ^ . The actual form of the equation in z will be i5'* + ~^"-+^;5'-' + &c. = 0. It is easy to see that the discriminant will appear in the denominator; and the coefficient of z*'^ will vanish, since, if <p be any function of the order w — 2, the sum of the results of substituting all the roots of U in-^ vanishes. In fact^ when the terms of this sum are brought to a common denominator, the numerator is the sum of <^a multiplied by the differences of all the roots except a, and this is a function of the order n^2 in a, which vanishes for w — 1 values of a, a = ^, a = 7, &c., and must therefore be identically nothing. In applying this method to the quintic (a;, 1)', Hermite substitutes where ^j, ^j, ^g, <^^ are four covariant cubics of the orders 3, 5, 7, 9 respectively in the coefficients. <f>^ is the canonizant. <l>^ is the covariant cubic of the fifth order, the Jacobian of 8 and T whose leading term or source^ whence all the other terms can be derived, is printed in full as (11) Art. 232 ; on inspection we see that this source vanishes if both a and b vanish; consequently, if the given quintic has two equal roots, their 256 HIGHER BINARY QUAKTICS. common valae satisfies this covariant We can form a coTariant cubic of the seventh order from ^, in the same way that ^, was formed from <t>^^ and by adding <f)^^ multiplied by / and a numerical coefficient, can obtain <f)^^ such that its source vanishes when a and b vanish ; and, in like manner (f)^ can be made to possess the same property. When this substitution is made, the coefficient of 2^ is a quadratic function of a, ^, 7, S. * Hermite finds for its actual value (a result which may be verified by working with the special form, note, p. 248), {Fa' + BKDay - D {F+ 10 JK) 7'} + D {K^ + 2F^8 where -^=9 (16ii— /jST), which vanishes when the quintic has two distinct pairs of equal roots. By breaking up into factors each of the parts into which this coefficient has been divided, the two linear relations between a, 7 ; ^8, 8, which will make it to vanish, can readily be obtained ; as also by another process which I shall not delay to explain. The discussion of -this coefficient is also the basis of Hermite's later investigations as to the criteria for reality of the roots. He avails himself of a principle of Jacobi's [Crelley vol. L.), that if a, /8, 7, &c. be the roots of a given equation, and if the quadratic function (t + au + a'v+...a"^wy + {t + fiu + I3'v + &c.)' + &c., be brought by real substitution to a sum of squares, the number of negative squares will be equal to the number of pairs of imaginary roots in the equation. Hermite shews, by an easy extension of this principle, that the number of pairs of imaginary roots of the quintic is found by ascertaining the number of negative squares, when the coefficient of s^ just written is resolved into a sum of squares. And since the same process is applicable to every equation whose degree is above the fourth, he concludes that the conditions for reality of roots in every equation above the fourth can be expressed by Invariants. 249. It does not enter into the plan of these Lessons to give an account of the researches to which the problem of BE80LVENT8 OF QUINTIC. 257 resolving the quintic has given rise.* The fallowing, however, finds a place here on account of its connection with the theory of invariants. Lagrange, as is well known, made the solution of a quintic to depend on the solution of a sextic ; and it can easily be proved that functions of five letters can be formed capable of rix values by transposition of letters. Let 12345 denote any cyclic function of the roots of a quintic; auch|. for example, as the product («-/3)«(|8-7r(7-Sr(8-e)''(e-a)', where evidently 23451 and 15432 would denote the same as 12345; then it can easily be seen that there can be written down in all twelve such cyclic functions. But, further, these distribute themselves into pairs ; and by so grouping them we can form a function capable of only six values; for instance, 12345+13524, 12435 + 14523, 13245+12534, 13425 + 14532, 14235 + 12543, 14325 + 13542. The actual formation of the sextic having these values for its roots is in most cases a work of extreme labour. M. Hermite, however, pointed out that when the function 12345 is the product of the squares of differences written above,t all the coefficients of the corre- sponding sextic are invariants, and that the calculation therefore is practicable. I have thought it desirable actually to form the equation, because, in the development of the theory of sextics, it will be necessary to ascertain the invariant characteristics of sextics whose solution depends on that of a quintic; and it may be useful to be in possession of more than one of the sextics which Earing out of the discussion of a quintic4 I take the simple example x*+ 2ma;y + asy*, of which, * Among the most remarkable of recent investigations in this subject is the application to it of the theory of elliptic functions 4>y M. Hermite and M. Sjconecker. t In the method of Messrs. Harley and Cockle, the function 12345 ia a^ + /8y + yi + ^« + «a, and the sextic chosen is that whose roots are 12345 — 13624, &G, This baa been calculated by Prof. Cayley {Philosophical TraruactimUf 1861, p. 263), and the result is very siniple, two terma of the sextic are wanting; bttt the coefEcieats axe not invariants. X The form arrived at by M. Kronecker and M. Biioedu is (x - a)» (x - 5a) + 10* (a? - a)» - c (a; - a) + 66^ - «? = 0. By the help of the f<Minul» giv^i further on^ the invanants oi this equation can be calculated, and a, i, e eliminated. , LL 256 HIGHER BINARY QUANTIOS. since two pairs of roots are equal with opposite signs^ the functions of the differences can easily be formed. I find then that the sextic is the product of «* + 2'(wi4 m') « + 2" (m^- 2m* + 5m'), by the square of «* + 2* (m"+ 3m) <+ 2' (^^4 5m* + 19m' ~ 25). But if we first multiply the quintic by fiye, its invariants are /= 2*m (5 + 3m«), J9 = 2". 5» (1 - m')', Z = 4m (5 - m')*. To avoid fractions I write J =^2 A, 2) = 2505, J'-2"i = 50(7; and then forming the sextic, and expressing its coefficients in terms of the invariants, I obtain «• 4 4^t' + (6^" - 25B) «• + (4^' + 2 C - 30^5) «• 4 <" M* 4 4-4 a- 17 A'B 4 ^f^S") 4- 1 {2A*0" 4A'B - 7BG 4 110^5") 4 C7* - 4^5(74 20 A*B\ which is a perfect square, as it ought to be, when D = 0.* • 250. M. Hermite has studied in detail the expression of the invariants in terms of the roots. He uses the equation trans- formed so as to want the first and last terms ; that is to saj, so that one root is and another infinite ; and the calculation is thus reduced to forming symmetric functions of the roots of a cubic. I had been led independently to try the same transformation on the problem discussed in the last Article, but found that, even when thus simplified, the problem remained a difficult one. It would be necessary to form for a cubic the sextic whose roots are the six values of a;(/3-7)«(a-7)\+/3V(a-/8)', and then to identify the result with combinations of the forms assumed by the invariants of the quintic when a and y vanisL M. Hermite expresses his own invariant / as follows. Let a, = (a-/3)(a-e)(S-7) + (a-7)(a-S)(/3-e), «, = (a-/3) (a-7) (^ - 8)4(a-8)(a- s) (/3- 7), a3»(a«/3)(a-8)(8-7) + (a-7)(a-£)(S-/8); the continued product of these is symmetrical with respect to all the roots except a ; and if we multiply this product by the similar products obtained for the other four roots we get /. ♦ Though the form with which I have worked is a special one, I bttlieve that the vesult is general ; because it seemed to me that the coefGicientB only admitted of bdog expressed in terms of the invariants in one way. CONCOMITANTS IN TERMS OF THE BOOTS. 259 These factors are of course the values of the determinants a', 2a, 1 i 2a - (7 + e) , a (7 + e) - 275 , &c., iSS, i8+8, 1 7S, 7 + e, 1 which express, p. 193, that one of the roots is self-conjugate of the involution determined by the other four, which is the case when / vanishes, as remarked. Art. 229. Determining the numerical constant by a special form, such as ax^+5exy*-\-Ji/'^=0j we find the product of these fifteen factors by a" to be 10"/. 251. From the roots of a quintic five sets of four can be formed by omitting each in turn, let Saj /8/3, Sly, S^, 8t denote the equi-anharmonic functions of these sets of four, see Art. 199 ; also Taj T^j Tyj Tdj Tt their harmonic functions ; it is easy to see by comparing terms in a simple case, for instance, for the quintic oa?* + lOcajy = 0, that we have in terms of the roots ^^^ 8^ 8a {X'ayy^8p{x-I3y)\8y [x-r^y)\ 8,[x-hy)\8, {x-tyf a 6000 r= Ta{x^ay]\ T^[x-Py)\ Ty{x-r^y)\ T^[x^Sy)\ T, [x-eyy - 1 00^= a^^S (a - iSy {x - 7^)' {x - Syf {x - sy)*. If we calculate for the quintic aod^ -f- lOdx^y* = 0, the Value of the ten terms S (a - ^8)* (7 - Sy [S - e)* (s - fif which we saw, Art. 241, can only be the quartinvariant, we find a*2 (a - By (7 - 8)* (S - e)" (e - 7)" = - 1250/. The function 2 (a - /3)' {0 - 7)' (7 - By (S - e)« (s - a)» containing twelve terms (Art. 248), is found to have precisely the same value. Similarly it may be noticed, for the covariant J?, note p. 240, a*S (a - ^y (7 - 8)* (B - e)» (e - 7)'' {x - ayy {x - I3yy = lOOOi), whose source is the function D written in full on page 230. The value of the discriminant B on the same page must be multiplied by 5" to become identical with a® times the product of the squares of the differences. The condition that four of the five roots may be equi- anharmonic will be that some one of 8aj 8^^ &c. vanish. Hence their product will be an invariant. We get by a special form a'Sa8^8y8i8^ = 20^ (J' - 3K}. 360 HIGHER BINARY QUANTICS. Similarlj for the harmonic condition, we get the invariant Again, the function 2 (a - /8)* (a — 7)* (S ~ e)* consists of six terms belonging specially to a, which we may denote bj JS^y and similarly for J9j3, &c. It is easy to see that, if we write a for a; : ^ in the above value for the covariant Sy the .right side becomes 2 Ha. Hence, the eliminant of 8 and u is the product of these five factors. But by Sylvester^s canonical form or otherwise, that eliminant is found to be 2JK'-9L-'J^j and by a special quintic such as (a? - 1)* {x -\-2)x*^ 0, we can find the value of the constant, thus 3a''HaBpHyHiH, = 10' {2JK- J' - 9i). Similarly we can see that / is the eliminant of T and u, for, the result of substituting a in the above value of T is six times the product of the factors o^, a,, a,. Again, by taking a special quintic of the form used in Art. 241, we find the constant which gives the symmetric function, a"2(a-/8)'(i8-7)'(7-a)"(e-ar(8-.i8r(e-7)*{S-ar(S-/8r(S-7)* « i . 5" (2"i - eT^H ^JD) = 2.5' {5.2'L - eT^' - 2V^).* 253. The Sextic. The investigations of Clebsch and Gordan show that, including the sextic itself, there are in all 26 forms. There are four independent invariants, which we shall call A^ By Cj Dy of the orders 2, 4, 6, 10 respectively; a fifth Hy of the order 15, is skew and its square a rational and integral function of the other four. There are six quadric covariants whose orders in the coefficients are respectively 3, 5, 7, 8, 10, 12 ; five quartics of orders 2, 4, 5, 7, 9 ; five sextics, orders 1, 3, 4 and two of the sixth; three octavics, orders 2, 3, 5; one decimic, order 4 ; and one duodecimic of order 3. The first invariant A is 12*, formed by the method of Art. 141, and for the general sextic is ag - 6J/*+ 15ce - lOt?. I have given (Art. 174) the canonical form of the sextic; but I believe it will be found in practice not less convenient to use the more general form ^ ■ - II I . I - ^ ■— M^^ " ♦ The equation determining the anharmonic ratios of the iroots has been given by Mr. M. J. M. Hill. Proc. Lond. Math, Soc,, vol. xiv. THE SEXTIC, 261 To this we sbould be led by the theory of two quintlcs, which cannot be more simply expressed than as each the sum of four fifth powers. For the form just given, the invariant A is, by proceeding as in Art. 223, found to be '2ab (12)*, or, in full, ab (12f + ac (13)* + ad (14)* + be (23)* + bd (24)* + cd (34)*. The Hessian of the sextic, 12*, is of the eighth degree, the g^eral coefficients being ac— i', 4(arf-Jc), 6a6+ 4Jc?— 10c*, 4a/+Ub€'-20cd^ a^+14J/+5ce -20c?',* &c.; and for my canonical form is Xabu*v* (12)'. The sextic has another covariant of the second order in the coefficients, with the variables in the fourth degree, viz. the 8 of the emanant quartic, which is for the canonical form '2abuV (12)*, the general coefficients being ae - 4id + 3(5*, 2a/— 6&6 + 4c(f, ay — 9ce + 8d**, &c. To these we add, the covariant sextic, of the third order, the T of the quartic emanant, which for the canonical form is 2aJc(12)' (23)* (31)' uVw\ and whose general coefficients are aee + 2bcd -ad"- eJ' - c*, 2ac/- 2ade - 2 jy + 2bce + 2bd^ - 2c*rf, acg + 2ac^- 306* - h'g - 2bc/+ ihde + 2c*6 - 3cd*, 2adg - 2aef- 2bcg + ^bdf- 2be^ - 20*/+ Gccfe - 4d"*, &c. Abo, the simplest quadricovariant Z, of the third order, 12* of 8 and u^ or 12* of u and H^ which for the canonical form is SaJc (23)^ (31)' (12)' «', and whose general coefficients are acg - Vg - Zadf+ Sbcf+ 2ae' - Jde - 3c'e + 2cc?* = 7^, adg-bcg- aef- Sbdf+ Oc'/ + 966* - 17ccfe + 8cP = 2Z„ aegr - 3Wy + 2c''g - a/' + 3 Je/- cdf- See"" + 2cPe = 7,. 253. We take for the invariant B that which has been called by Sylvester the catalecticdnt^ which expresses the condition that the sextic should be reducible to the sum of three sixth powers, and is (Art. 171) the determinant a, 5, c, d i, c, dy e c, d, c, / rf, 6, /, g * It seems unnecessary to write the terms which follow from symmetry. 262 HIGHER BINARY QCANTtCS. This expanded is - ibd^f-\r ibde\- (?g + 2c'^+ cV - Zcd^e + cP. If now we form the quadrinvariant of the Hessian, we find it proportional to -4' + 300-B; if that of the covariant /S, we find -4' — 365 ; and if we operate on the sextic with the covariant jT, we get B. Applying this last process then to the canonical form, we get the value of jB, iJbcd (12)* (23)" (34)* (41)* (13)* (24)', which vanishes, as it ought, if any of the quantities a, &, c, i vanishes, or if any two of the four functions u, v, u?, z become identical. 254. We take for the form of the fundamental sextinvariant 6\ that which involves no power higher than the second of the leading coefficient a, and which for the general form is C» a*tf / - ^a^defg + 4a*(^' + 4a'eV - 3a*ey * - ^ahcdg^ + ISoicej^ - 12aJc/' + Uabd^fg - ISabde^'g + 6a5ey+ 4acy - 2lacVg - ISac^d/g + SOad'ef + dS.accPeg - 12ac(P/* - 42acefe'/ + 12ace* - 20acfg + 24(wf €/- Sad'e'' + 4J'd^* - UV'e/g + 86'/' - 3JV/ + 306»C6'^ - 24JV" - l^i'cTeg - 24J*e?/* + 606Wey- 276V + 66c'/^ - 4260'% + GOic'rf/^' - 306cV/ + 2Abc<rg - 846a?V+ 666a?e' + 246^*/- 246(f 6* + 12c*^ -27cy*-8c'(f^+66c'6/€/-8c'6'-24c'd'/-39c'(f6*+36c^e-8d*. In terms of these we can express the other invariants of the sixth order. Thus, the cubinvariant of the covariant quartic is -4'— 108^4-5— 54(7; the cubinvariant of the Hessian, p. 141, is 3A^-100AB-\'2750C] the discriminant of the quadratic I is 4(?^Z, — Zj^) = 4-4jB + 3(7; and the quadrinvariant of the sextic covariant is 2AB— (7. The last-named invariant can be easily calculated in the case of the canonical form. We have to operate with SaJc (12)'(23)'(31)VvV on itself. Now if we operate with uVz^ on v^vV the result is proportional to (12)W^, where M and N have the same meaning as in Art. 223; and if with uVw^ on itself the result is -(12)' (23)* (31)*. Hence we get for the invariant in question 2a'6V (12)^ (23)' (31)« - 2abcd^ab {i2yM^N\ DISC?RIMINANT OF A SEXTIC. 263 255. If a, bj c all vanish, the invariants Aj Bj G become respectively — lOcP, d^^ — 8t?. Hence, when the sextic has as factor a perfect cube, the conditions must be fulfilled -4'= lOO-B, 445=5(7, AG=^SOB\ If we make a, J, /, g all =0, the invariants become consequently when the sextic has two square factors, in addition to the discriminant vanishing, the condition must be satisfied, {A* - dOO AB + 250 (7)' = 5 (4' - 1005)". 256. If we make ft, rf,/= in the equation, the discriminant will be ag multiplied by the square of the discriminant of (a, 5c, 5e, fl^Ja?, y)* ; and if all the terms vanish but a, rf, g^ the discriminant will be a'^' multiplied by the cube of the discrimi- nant of (a, lOdj gJixj y)\ Knowing these terms in the discriminant, the rest can be calculated by means of the differential equation. The resulting value of A is ay + 43500 a'bdeTg' + 30000 aW -30 a'b/g' -7500 a'bd/'g - 330000 a'^efg' -300 a*ceg* + 57000 a*h,yg* + 50000 a'^rg + 375 a^cfg' - 97500 a'bey'g + 250000 a'd'e'g' -300 «V9* + 37500 cfbef + 675000 a'd'eyg + 3000 a*de/f + 1000 a'cY - 375000 a'd'ef* -2500 a*d/y - 27000 aVdfg' - 900000 a'deyg + 1000 aV/ + 18750 aVeV + 500000 a*dey' -7500 a^eyy + 16875 a'cV/y + 250000 aVg + 9375 aVV -9375 a'cTg - 150000 aVf* -3125 ay -7500 cfccPe^ -2500 a*b*dg* + 375 a'b*eg* + 127500 a'ccffg' + 750 a'bYg' -15 a%Tsf + 30000 a'cd^M -410 a'byy + 3000 a'bcdg* - 412500 a'cdefg -7500 a*bVg* -5550 a'bce/if + 187500 aWf + 43500 a'Vcd/g' + 750 a'bc/Y -150000 a'ceY + 16875 a'bWg' -4800 a'bdyg' + 412500 a'ce'/V - 64675 a*bW/Y - 27000 a'bdeY - 187500 aWf* + 25500 aVc/*9 264 ^HIGHER -BINARY QUAISTTZCS. 4 127500 - 171300 -346500 + 616500 -240000 + 7500 - 23250 + 11250 + 57000 + 30000 - 346500 - 596250 + 1222500 - 506250 -330000 + 1590000 - 330000 + 750000 -3172500 + 1537500 + 375000 - 225000 + 780000 - 1350000 - 2190000 + 1200000 + 4650000 - 2550000 - 1500000 + 900000 - 150000 + 7500 + 250000 + 750000 + 37500 a^Vdeyg" cfVdePg a^bccPg^ a^bcd^efg^ a^bcdeY a^bcdeYg €^hce\fg cfhcip a^hd'fg^ o^hd^efg a^hd^f a^hd^e'/g c^hdig d'hdeT V 4 8 aceg a\^d^g^ a^c^defg^ a'c'd/g + 1062500 - 2821875 + 1265625 - 1350000 - 3562500 + 4725000 - 2062500 + 4875000 - 2625000 - 1875000 + 1125000 + 3750000 -300000 -9750000 + 5250000 + 3750000 -2250000 -1000000 + 3000000 -1600000 - 1250000 + 750000 + 9375 -7500 -9375 + 25500 - 11520 - 97500 - 412500 + 616500 + 1222500 - 2197800 + 864000 + 50000 - 330000 a^^d^ff cf^d^eY a*<?dPeJ^g c^c^d^f" a\'dejg a\^dey^ a\*e^g cfcd^eg* dfc^f^g cfcd^^fg dfod^iP a'd^g^ a^d^efg a'd^r a^d^e^g a^d^ey^ ah^cg^ aVd/g"^ aVey abV/g^ aVcdeg^ ahW/g' ob^^c^rg a}?d^g^ aVd'efg' + 83200 cd?d^f*g + 37500 aJ'&y + 511500 aVde]Pg -288000 aVdef^ -202500 cd^e^g + 121500 aJ'ey* + 412500 aJ*cV - 23250 ei?^ff + 675000 oJVtZy - 3172500 dh^&deff + 511500 ab^^dj^g - 2821875 oftVey + 7633125 aV'cVfg - 3442500 oftVtf/'* - 21%000 ai*cdy/ + 4725000 aJ*ce?ey + 6030000 ab^cd^efg - 3360000 11^0?/* - 15337500a5*crfe^ + 83^2500 ah^tsd^P + 5062500 ah\ig - 3037500 ah^c^P -300000 aVd^egt + 900000 od>^d^eJg -480000 ah'dPef -375000 aft^tTeV + 225000 €d?d^ip -900000 cMd^ + 375000 ahc^eff -202500 ahcPg + 4650000 Qbc^JPfg* + 4875000 oftc'&y - 15337500aW&/^j + 7087500 alH?dp + 843750 abcVJg INVARIANT OF TENTH ORDER. 265 -506250 aJcVy - 9750000 ahi^JPeg' + 900000 abc^cPf^g + 24750000aJc'(re*/'y - 13350000aic*d*e/' - 9375000 abc^de^g -f 5625000 cibifdip + 3000000 ahc^g'' - 9000000 abc^efg + 4800000 ahcd^f^ + 3750000 abccP^g - 2250000 abi^e^p + 250000 ac*^ - 1500000 ao^df^ - 1875000 ac*ey + 5062500 ac^f^g - 2278125 a&f" + 3750000 ac^J^eg^ -375000 ac'^d^fg - 9375000 axfd^fg + 5062500 ac*(fe;r' + 3515625 ac*eV -• 2109375 ocV/* - 1250000 a^d^g^ + 3750000 add^efg - 2000000 a^d^f - 1562500 ac'i*eV + 937500 acVe"/' - 3125 jy + 37500 V.i^g^ + 187500 6'%" -240000 Vdfg^ -506250 iV^ + 864000 i'e/'y -331776 £•/• -187500 5Ve^" + 11250 6V/y -375000 6*cdf/ + 1 537500 Vcdefg" -288000 h^cdfg + 1265625 J*cey - 3442500 iV/ V + 1555200 i*ce/* + 1200000 i*dy/ - 2062500 6Vcy - 3360000 Vd^efg + 1843200 6*(f/* + 7087500 6W;& - 3888000 6W/' - 2278125 6V^ + 1366875 6V/« + 500000 iVi/ -225000 6Ve/y + 121500 6V/V - 2550000 JV(%r« - 2625000 JViey + 8392500 Vc^defg -506250 VcVfg + 303750 J'cV/' + 5250000 Vcd^eg^ -480000 VccPfg - 13350000J"cd*e!/57 + 7200000 6'ci*e/" + 5062500 Vcde^g - 3037600 Ved^r - 1600000 i'(f/ + 4800000 6V^ - 2560000 yc?y' - 2000000 i'i'eV + 1200000 i'i'ey" - 150000 J"cy + 900000 Vc^d/g" + 1125000 6Vey - 3037500 Vc^efg + 1366875 JV/* - 2250000 Vc^di'eg' + 225000 V^d^fg + 5625000 V^d^fg - 3037500 iV&/" - 2109375 6VeV + 1265625 iVe'/" + 750000 JVrf*/ - 2250000 Vc^d^efg + 1200000 JVcP/' + 937500 6V<feV -562500 JV(fey - 3888000 V^df" 257. If it be required to determine a cubic whoae covariant 12" with a sextic vanishes identically, it will be found that the problem furnishes linear equations enough to eliminate all coefficients of the cubic, and the condition to be satisfied is, that the determinant B of the sextic vanish. But if the problem were to determine a quartic whose 12' with the sextic should MM 266 HIGHER BINABT QUANTICS. Tanish identically; though there are enough linear equations to give a determinant condition among the coefficients of the sextic, this will be found skew symmetrical of the fifth order, and therefore the condition is identically satisfied. Hence, this problem admits of solution generally. In fact, it is easy to verify that if we write the quartic /S, which we shall now call t = [ae - ibd+ 3c') x*+2 {af- She + 2cd) o^y + (a^-9ce+8ef )ajy +&c = i^qS" + ^i^y + 6*,jcy + 4*,ajy' + «y, then a*g — 36*, + 3ci, - di^ = 0, &c., thus it satisfies the conditions of the question.^ This quartic covariant furnishes to the system of the sextic besides its invariants its Hessian and its sextic covariant. The determinant value of B^ treated by the rule of Art. 219&, at once gives the invariant u4* - 365 = 12 ( V4 - 4ij*, + 30 = 1 2/„ as we shall write for brevity. If the coefficients of the Hessian of i be written out, we find that its source satisfies the relation Writing out the corresponding equations for the other coeffi- cients, multiplying the first by i^ the second by —4*,, and so on, and adding, also calling the invariant III -i- 2l I I — A^*— tt'— ^'"S-Z^. *0*^1*^4 • *1*^1 8 8 *4 1 8 SJ * It should hare been noticed that if in a skew symmetrical determinant of odd order the constituents in any row or column be replaced by arbitrary quantities, the determinant formed is a linear function of the binary determinants formed from the constituents of the replaced and its conjugate column or row. And again, if in a skew symmetrical determinant of even order all the constituents in any row or column be replaced by arbitrary quantities, the value of the determinant formed is the product of two linear functions of the constituents of the replaced and its conjugate column or row. Thus the value of the determinant , o , - 3* , 3tf , - ^ , 6o — o , , Be f —Sdf 3e, — ibi 36, - 6c , , 6e , - 3/, eb^ - 3c, 8rf, - 6e , , g, - 46, dy -3e, 3/, ^gy 0, b, enables us to write down in a determinant of the fifth order the product of the covariant i by any one of its coefficients. SYSTEM OP QUABTIC COVARIANT. 267 *re get SG/, + 2-47, = 12 [IJ^ - \^) = 3 {AAB + 3 (7), Art. 253, whence 108^45+ 54(7- ^' = 216/3. The relations jast employed show that the Hessian of t may be replaced in the system of the sextic by the result of operating with I on the sextic, and the sextic covariant of i by the result of operating with I on the Jacobian of the sextic and t. The same relations being employed to determine the quadrin- variant of the function (aZ, — 2J?j + cZJ a?* + &c., lead to the equation = I (27/ + 3^7,) = i {2A'B + 3ul C7 + 72B»). 258. It should have been mentioned in dealing with the system of a quartic and a quadric that if we call the quadric covariants at the end of p. 203 respectively ajaj' + 2)8^a?y + 7y, aV + 2/3'a:y + 7'/ , and operate with the former on the quartic a new quadric covariant a^x* + 2l3^xy + 7^* is found, if we operate with this on the quartic we get another a^x^ -{- ^fi^xy + 73y", and so on. From the system of equations thus derived Oj =07- ibfi + ca, a, =a7j-2&)8j + ca^, a, = a7, - 2 J^, + ca„ )8j = J7-2c^+rfa, /3, = J7j-2c;3^+cZa„ /3, = J7, - 2ci8, + rfa„ 7i = C7- 2rf)8+ 6a, 7, = <J7j - 2cZ)8j + ettj, 73 =07,-2^7,+ «a„ It can easily be seen that each covariant admits of linear ex- pression by two preceding it in the series, the values being a, = Sa^ + 2 Ta^ a^ = /8a, + 2 jTa^, &c. Moreover, these equations show that a7j — 2/9/8, + 7a3 = 2 [a^i^ — /8j'), and when we write 8oL^ + 2 jTa = a7, - 26/S, + ca„ &c., we see that a,7, - 2/3,/S, + 7,a, = 8 {ay, - 2^8^, + 7a,) + 4 r(a7 - ^), «,7, -^:^S (a,7, - ^:) + T[ay, - 2/8;3, + ya,). In the notation of Art. 212, 4 (a7 - /S') = 2', a7j- 2/3^^+ 7ai= «^, ^i7i — ^1* = 2 + ;S (a7 — /3*), thus the successive invariants of the quadrics are expressed in terms of these five invariants, and as the skew invariant difi^ers only by a factor from its square can be similarly expressed. «j /S, 7 268 HiaKEB BINABT QUANTICS. It may also be noticed as regards the eovariant fi that we find 8fl^,= 2/Sot + a^; and Ukewise for the aeries of derivatives according to the law corresponding to ti from the qnartic, a'" = 3iSV + 2 (542^ - S^) a, &c. Also ot" + iSx' = 2iS'a + ISJa^. 259. Now when we take as qnartic the eovariant t of a sextie and take its eovariant I as quadric, we derive a new qnadric m by operating with I on i, another n by operating with m on /, and by what we have jast seen, the further qnadric covariants thus derived are reducible. These three thus give rise to a complete qnadric system of which we proceed to consider the invariants. The relations between the coefficients of % and of its Hessian used in Art. 257 give 12 [a (V4- */) - 2 J {*,*, - Va) + c(V* +2v. - 3^/) - %d (v.- vJ 6(i,7,-2y,+ *,y = m., tn^f m, having similar expressions to that for m^ By the same relationS| determining the result of operating with the square of I on the Hessian of /, we find this invariant linearly related to the result of operating on the sextie with the cube of ?, al^ — ^hl^\ + &c. = D^, and to the last invariant written in Art. 257 ; but it is also obviously so related to the discriminants of I and of m. We find as the result of the comparison what we shall call ♦ In full, the value of m^ — aHg^ - 6a^dfg + %aH^g - ZaHp - oft^* + ^obefg + llcu^eg - ^^ilbd/tg - ^acd'^g + ^^abdp - 45acy« - ISodey + ISacdef- 48ady- 36ace» + 28ai2e2 - 9b^ceg + 7262^^ - 96*c% + 36c*g - 14Ah^def+ lOSbd^f + 96bcdy- 72c»df+ SWe^ - 1266c(fe« - 27c*d^ + 16W»« + 96c2rf2« - 32<jrf*. t It was mentioned in the second edition that instead of the discriminant A m may use another invariant of the tenth order i), in which no higher power *^*^ the fourth of the extreme coefficients a, g appears, and which does not contain ^e product a*g*. The quantity multiplying a* v^i D ia (^ ^f^)\ <uid the relation INVARIANT OF FIFTEENTH OfiDER. 269 We can now, writing 6 (t^m, - 2c^m^ + ^,wij « w^, &c., express the remaining invariants of the quadratics by means of the invariants -4, /,, /,, J7. In fact, using last article, we have fw.n, - 2w,n, + m,w, = 36 {/,(7,7n, - 2Z,m, + ?,^ J + 24/^ (Z,Z, - Z/)}, «o^ - V = 36 {/, [m^m^ - »i^«) + 6/, [l^m^ - 27,m, + Z,m,)}, and we have ab-eady at the end of Article 257, l,m^ - 2\m, + l^m^ = 4 (2// + 3^/,). We have, of course, also the skew invariant of the system of quadrics which must be to a numerical factor the E of the sextie, and its square will be expressed by the other invariants in the usual manner by putting in their values in the expanded formula for m„ ntif m. •01 t -2/„ I. *ii '4 "if ""Si*!, «• 4 (2/j« + 8^/,), 2iy 2iy, 28S (/,» + 2AI^t + 9/,«) 288 (/j» + 2^,7, + 9/,«), 72 (4D' + 48V/, + 72^/,«) moreover, the identical relation between the quadrics may be written down by equating this determinant bordered with them to zero. i (J/, + 18/,), 4 (2/,« + 8^,), 2D', 260. By means of the differential equation I calculated the invariant E. Its value was given at length in the second edition, where it • occupied thirteen pages, but I have not thought it worth while to reprint so long a formula. The terms containing the highest power of a are whence making all coefficients vanish but a and y the deter- minant at end of last article is found = 2 {27 Ey. ^^^——^•^^ III , connecting A and i) is A = 4* - 375-4»J5 - 625^«C + 3126i). The value of D waa there given at length, but I have not thought necessary to reprint it. Taking either of its special values in the cases mentioned at the beginning of Art. 256, D^ is fomi<| connected with it by the equation i^i = i> + 65 (3(7 + 2A£) : whence we find 9^ - 3126J)' = SS-Ll* - 12000.42 (^/jj ^ 5/^) ^ 75000/j (J/, + 6/3). 270 HIGHEB BINARY QUANTICS. The expression for E in terms of the other inyariants may be otherwise found from the following considerations: If in the sextie J, dj f vanish, E necessarily vanishes. For, since the weight of E is fortv-five (Art 143), the weight of some one of the constituent coefficients in each term must be expressed bj an odd number; and when in the sextie we make all the terms vanish whose weight is odd, E vanishes. £=0 is therefore the condition that the roots of the sextie should form a system in involution. If then we make 5, ^,/= in A^ B^ (7, A and eliminate a, c, e, g from the results, the relation thus obtained between A^ B^ (7, A must be satisfied when E vanishes, and must therefore contain it as a factor. From what has been just remarked it follows that the expression for E in terms of the roots is the product of the fifteen determinants of the form 1, a-f-iS, a/8 1, 7+S, 7S 1, e + ^, e^ or of the fifteen factors (Ex. 7, p. 25) (a-S)(7-0)(e-/S) + (/S-7){S-e)(*-^ If we write ag^\ ce = /A^ ae^'\'gc* = Vj the values of the invariants got by making J, rf,y=0, may be written A = \ + 15/i, £ = X/Ll + /Lt* — v, C = - 24X/U,' - 8/i' + 4 (X, + 3/i) v, A = \ {\' - 160\fA - 1875/a' + bOOvy. Eliminating v in the first place, the last two equations become C=4/i(\-/i)'-4(\+3/i)5, A=\(\'+350\/i-1375/u,'-5005)*. Then eliminating jn by the help of the first equation, we get 1024V-1152V^+ (132^'-108005)\+3375 (7+2700^5- 4^' =0, \ (256X.' - 320^\ + 55A' -f 4500B)* - A = 0. The resultant of these two equations is of the thirtieth degree in the coefficients; and therefore, from what we have seen, can only differ by a constant multiplier from E\ SKEW INVARIANT OP SEXTIC. 271 261. Bj Art. 234, when the discriminant is negative the sextic has either six or two real roots ; and when it is positive, lias either foar or none. We can readily anticipate that the discussion of this expression for E is likely to lead to the same results in affording criteria for further distinguishing these cases^ as the corresponding discussion of the expression G in the case of the quintic. Analogy also leads us to expect that what will be important to examine will be the result of making A = in the expression for E. Now, although the calculation of this general expression for E may be a little laborious, that part of it which is independent of A is easily obtained. It will evidently be the product of 8375(7 + 2700^5-4^" by the square of the resultant of the cubic and of the quadratic 256V - 320^X, + 55^1* + 4500S. And again, analogy leads us to believe that the first of these factors is not important in the question of the criteria for real roots, and that it is the square factor alone which needs to be attended to. . The result I find is that, writing for convenience B' for lOOi/, C for 125(7, the quantity squared differs only by a constant multiplier from 44* - 194*B' - AdA'B" - 4.A* G* - 805" + h2AR 0' - 4 G'\ Analogy then leads me to suppose that the criteria for the number of real roots of a sextic depend on the signs of this quantity, and of A' - 1005, A^ - 125 (7, {A* - 30045 + 250 (7)* - 5 {A* - 1005)', which, as we saw, vanish when three roots are all equal. 262. If we now resume, from Art. 223, the consideration of the system of two quartics u = ax^ + ^hx^y + 6ca;'y* + A:dxy^ + ey^ , V = a V + 4 J V^ + GcVy* + ^^ocy^ + eV ; and, as in Art. 216, write their Jacobian or functional deter- minant, which is in full, (a&O aj' + 3 (ac') x'y + 3 [{ad') + 2 [he')] x'f -f {[ae') + 8 [hd')] xY + 3{(JeO + 2(ce?)}xy + 3(c6')^/+(^^0/ = a,a5* + Qa.x'^y + 15a^xY + 20a^xy + Ua^xY + Qa^xy^ + ay, 272 HIOHEB BINART QUA.NTICS. and their quadric oombinant of the same order in the coefficients {{ad^ - 3 {be')} x' + {(«?') - 2 (W)} X2f + {{he'] - 3 («f)l 3^ we express as follows all the determinants of the second order employed in Arts. 220-1 in calcolating the combinant invariants, (oJ')= a„ (a0=«2a„ (arf') = 3a, + |p., (6c') = o,-iA. («')-2a., {ae')=>ia, + fp„ {hd') = 2a,-ip„ {de')= a„ (Je')=3«4 + IP,. K)= ««-*?.• Between these determinants we have five identical relations of the form {ab') {cd) + {ac') {db') + {ad') {be') = 0, but of which anj two result from the other three, and these introduce the simplest quartic covariant of our sextic Jacobian. In fact, writing as in Art. 257, «o«4 - *«i«8 + 3«,' = t„ Ot^s - 3«,«4 + 2a,a, = 2*„ &c,, the identities are "oPu- ^"iPi + «.Po = 5», - ip', «i P, - ^",Pi + o.Po = 5*1 - IPo Pu a,p, - 2a,p^ + a, p, = 5t, - .^ {p^p^ + 2p,»), «./'. - 2a«?i + a./*. = 5t, - |^,p„ «4Pi - ^%Pi + a,P. = 5t« - f ft'- Now If, in order to combine these, we write, as in Art. 252, «o»4 - *«!*» + Wi - ^a*. + Vo = 2^0, «i*4 - *«>*» + 6 Vi - *«4*i + a«*. = 2?„ also calling «o«» - 6«,«. + 15a,«4 - J O^.' = ^.. i'oft --Pi' = ^. «oft* - 4«, Aft + 2a, (ftft + 2;?,') - 4a, p,p, + a^y," = [a^jp,*], we find 2 {t„;>, - 2t,^. + t,p„ + |^,pj = 5 . 2 . Z, - | [o,^,»], Kft'] = 5 (f„ft - 2t, p, + t,p,) - -^Pp^, ^2I,-J,K^K', JAOOBIAK OF TWO QXTABTIOS. 273 from which eliminating [a, p,'], also writing down similar relations for the following coefficients and patting for brevity K^ M, - ^Pj we get ^J'. - 2i,pj + v^p, + Kp,^ |Z,. It will be observed that the ^ and I coefficients depend solely on the coefficients a, hence these equations enable as to determine the p coefficients by means of those of the sextic Jacobian, In fact, they enable os to solve linearly for the p coefficients in terms of these quantities and P, and when this is done to form from them the value of P. Thus the notation of Arts. 256, 258 gives us the determinant we find incidentally 50/, + 4ZP= 15 (ZoP,-2Zjp, + Z,Po)» ^^^ finally to determine P= V (-^a "" ^^) arrive at the quintic* 6K' - A^K' - 10I,K' + 2 (^,7, + 15/,) K' - 8^,/,ir Hence we see that as the Jacobian is a sextic of full generality, it is possible to express any given sextic as the functional deter* minant of two quartics in five ways; when any root of this quintic is employed, a quadric p is linearly found by means of the given sextic by the above equations, or we have definitely in terms of the covariants Z, m, n of the sextic: f (27, - 7,S:+ jSr'jp = (iT - 7,) Z - JJiw + ^n, and hence the values of the determinants [acTjj &c., are found. * In prepaiing the foUowing sketch of Dr. BriU's paper, Math, Annahn xx., p. 830, I had printed thus far without seemg the "M^moire sur les faisceauz de formes biiiaires ayant une mSme Jaoobienne," by M. Gyparissos Stephanos, in Tome xxvii. of the Savantt Etrangers of the Acad^mie des Sdencea, 1888. M. Stephanos obtains this quintic, but with a slightly difiGerent notation, p. 78. la the preliminary notice in the Comptes Rendut, 12 Dec., 1881, it la girem incorrectly. NN 274 HIGHER Bin ART QUANTICS. 263. From the system of three quartlcs we can obtain a similar theory. In fact, they afford the sextic covariant aa^ + 2/% + yy*, /3!B* + lyxy + ^y», yo? + 2^aJy + ^ a^ + 2/3^ + yy,"/3;^« + 2y^ + ^y, y/c« + 2^^+«y a'a:« + 2/8'ary + yy, /9'a:* + 2y'ay + d'y», y'a:* + 2«'ajy + «y which, using (0^7) to denote the determinant {oL^fi^ &c., we may write = (ai87)aj'+ 2(a/8S)a:V+[(a/Se)+ 3(a7S)]ajy+2[(a7e)+ 2{fiyS)y/ + [(aSe) + 3 (/97e) J a^y +2 (/8Se) rry* + (7&) y* «= 5^a;' + 6J,a^y + 15J,a;y + 20J,a;y + 15J^a;y + GJ^ay* + by. Introducing the present notation, we find the expansion (p. 3]) of the determmant o» *, <?, <^ e «', *', c, '^t. e' <h A y» *, «/ «.» /3., %. *., «/ «', /3', y', «', «' = a.5, - 6a,5, + 15aA - 20«,5, + 15rt,J, - 6a,J, + a,J. - ii^. {(«&) - 2 (/87e)) + ipa(a7s) - 8 (/37S)} - iij, {(a/3e) - 2 (078)). This introduces a new combinant.of the three quartics, •which we shall write Jjic* + 2g'ja;y + j^' = [(a^Se) - 2 (078)] a;' + [(«7e) - 8 {^yB)] xy + [(aSe) - 2 (/37e)] y* ; by whose coefficients along with those of h we can express ail the determinants* * It is useful to haye both notations : the identities following may be written by letting X and y go through all pairs of values of a, /3, y, 5, t in the matrix '/» y, %» y'» (a^yy) and are /SgCj — /SjOj + /34O0 = 0, a^Oi - Ooag - ^a, = 0, «2«6 - «1«6 - ^4«3 = 0, /^«« - /5s«5 + ft«4 = <>• FUNCTIONAL DETEEMINANTS. !§75 Among these there are five identities, as in Art. 28, three of which only are independent, and which lead to the system of equations of the type Jo?. - 2&,?. + J,?„ = k; - 10 (6A - ^\K + 3V), whence the determination of the covariant q from b^ by aid of which any given sextic may be identified with the functional determinant of three quartics, follows immediately as in last article. It is to be observed that the duality, of which these two articles furnish an example, is general; that if we have a system of p independent quantics of the n^ degree, n not less than p^ they have a functional determinant of the degree {n- p-\-\)p^ which is a quantic of full generality of that degree, and that any quantic of the degree [n-p-\'\)p may be identified, by adjoining to it a suitable irrationality, either with the functional determinant of p quantics of the n^ degree or with the functional determinant of n-p-\-l quantics of the n^ degree. The irrationality in the present case for either form is the root of a quintic, by means of which the quadratic covariant is linearly determined. Moreover, the system of combinants of the p quantics of the n^ order, for n not less than p^ is in number and form identical with that of n ^p + 1 quantics of the same order. 264. If the quartic XU+fiV+vW break up into the quad* ratic factors on comparing coefficients, we can find easily that the second factor may be determined, in general, by the first, as o» «/» «', '•o* 0, 4A 4/3„ 4^, 2n, rw By, 6%, 6/, »•» 2ri, n 4*, 4a^j 4i', 0, rt, 2r» «» «.» «', 0, 0, »•« 0, 0, 0, aj«, xy, y^ whence this particular value of \U-\- fiV+vW is found by replacing the bottom row.by J7, F, W^ 0, 0, 0, and accordingly \: fA : V are determined by r^ : r^ : r^. 276 HIGHEB BINABT QUANTICS. The second quadratic factor becomes indeterminate if the former make all determinants vanish in the matrix «h «*f «', r.f 0, 4/3, 4A, 4^', 2r„ »••, 6y, 6y« 6y', »•» 2r„ »•• id, 4*., 4a', 0, »•» 2r, •» «.» «'. 0, 0, '-.^ the former factor accordingly fails to determine the ratios \:fi:v. The equations which make this matrix vanish maj be written V«S4*iV,+26,(r^j+2ri«)-4*^,ro+Vo'=f(»'t^«-n')?o-'Aj(»*o?2-2r,7i+»V?o)»'t» Vi*-4*«^«^i + 2*4 (»'o»'a + 2ri«) -46,r,ro+ Vo* = I (Va-^-t*) ft - A (Mr-2riJ,+rigJ r* or in the form ^r,\ - 8 v,a, + r,r, (6a, - /8,) + 4r,»/9, - 3r,r,/9, + r,«/9, = 0, ^^-o'ai - Vi (6«, + ^,) + 12 V,a,+ 3r,«;3, - r/, (6a, + /SJ + 2r/a. = 0, and from these either of the ratios r^ir^i r,, may be eliminated dialyticallj and the other will be found bj a cubic. A root of this cubic being employed, the former ratio will be found linearly. Thus three quadratics u^j u^^ u, can be found, and we may assume, as the basis* of the system of quartics in all matters relating to combinants, the products u^u^^ u^u^^ u^u^. If, further, we suppose a linear transformation effected, making the independent variables the factors of te^, so that u^^xy and r^ = 0, r, = 0, we must have in the equations determining these coefficients /8, = 0, 13^=^ Oj ^, = 0, whence follow, by the identities of Art. 263 (note) that -^ = ?? = ?^ . Thus, if we a* a. «• make by suitable determination unity the common value of these quotients, we have by a linear transformation, depending on the solution of the quintic and on that of the cubic, arrived at the reduction of the general sextic to Dr. BrllPs canonical form a;' + 2pa;' + 3 joj* + 4ric' + 3aj" + 2px + J = . ♦ CJompare the Notice of Dr. Brill's paper in the FortsehritU der Maihmatikf 1882, by Dr. W. F. Meyer ; also his Jpolaritdt und rationale Curven, p. 305, dkc. CANONICAL FORMS OF SEXTiC. •277 . $65. When the quartic \U'\- fiV+vW la the square of s^x* + 28jXy + s^y*j we must have the matrix 8y, . 3y„ 3y', «^ + 2^ !but this is equivalent, by the identities of note Art. 263, to only two equations ; for example, we may write the two aoVi-ia,(V, + 20+VA-V«* = ^» Eliminating s^ : s^ between these, we get a biquadratic in 8^1 s^j and the second equation connects a single value of s^ : s^ with each of its roots. Thus four quadratics are found to solve the question: calling them 0,, 0„ ^3, ^^, we may take the 8quaj*es of three of them as basis of the system as regards combinant properties. It is obvious that, besides the linear relation which holds between any four quadratic functions, there must exist also a linear relation between the squares If ^j — ojy , so that s^^O^ *t = ^ 5 *t® vanishing of the above Bnatrix independently of s^ requires that a, = 0, a, = 0, /3, = 0, /3, = 0. . lo this way we are led, by employing a root of the quintic and ^ root of the biquadratic, to reduce by linear transformation the general sextic to the following canonical form, given both by M. Stephanos and Dr. Brill, x^ + ax^ + bx* +0x^ + 1=^0. 266. When, by the methods just explained, the sextic is identified with the Jacobian of a system of quartics, its dis* criminant breaks up into factors. For the case of two quartics these are (Art. 180) their resultant B and the invariant called D (Art. 221), which vanishes if u + Xv admit of a cubic factor. The resultant was already given (p. 220), but we may write it in the notation of Art. 261 as 12 = 12* 278 HIUUER BINARY QUANTICS. When we redace this determinant, as can be done very simply by the rule of Art. 219&, we find 3JJ = 4* {75 JSr* - 20A,K+ 8^/ - 125/J. The invariant D will be found in the same notation by expressing that we have simultaneously ajjB* + (4a, + \'p^ x^y + (6a, + fi?J «?'/+ (4a^+ ii?,) a?y' + ay = 0, (a,-i2?ja* + (4a,-|pja;'y+(6a,-i;>,)a;'y'+4a^»+ay = 0, and at once this is written down as a determinant of the sixth order. Dr. Brill shows that the discriminant breaks up Into two corresponding factors for the functional determinant of p quantics, and in particular determines their values for three quartics. For his canonical form (Art. 264) it breaks up into a factor of the sixth degree, »^!Z*+'?(^P'-2)(2*+l)+(ifr/-2)y-iSr(2'-l)y+i?(2*-l)*=iVC^, and its other factor is the product of four linear factors 2>^ = (jp - r)* + (j - 1)*, J9j=^ + r + j+l, J9^=:-jp-r+ j+1. From these, taking p, ?, r as rectangular coordinates, after the manner of Sylvester (p. 242), and considering how space is parted by the surfaces (7=0, D^ = 0^ ■^i = ^> t® investigates the number of real and imaginary roots for real values of ^, j, r. We omit the discussion, however, as well as the geometrical developments given in the M^moire of M. Stephanos, and conclude with a few miscellaneous examples on the subjects of this Lesson. Ex. 1. If the three quartics in Art. 263 have a common factor, we may equate them to zero, and, multiplying them first by x and then by y, eliminate a^, 2^, Ac Hence we get a determinant A = o, 4/3, 6y, 4d, «, «/, 4^., 6y„ 4a„ e„ o', 4^, 6y', 4^*, «', 0, a, 4^, 6y, 4a, t 0, a„ 4/3„ 6y„ 4d„ i, 0, o', 4)3', 6y', 4^', c' which we can write in either notation as follows A = 4 {144oo06 - 64oi05 + 24 (a^/S^ + a^^ + 4^Si - W} = 48 {8 (*/, - 6JA + 15^4 - lOV) - i (Mt - 2'i')l. exampi.es. 279 fix. 2. If we take as the three qnartics in Art. 263, the simplest quartic dovariants of the qnartics in Art. 262, viz. H=aa^ +...= {ac - fc*) a;* +..., H, = a/c* +...= (flk?' + ca'- 2W) ai* +..., E' =4t'7^ +...= (oV - 6'2) a?* +..., we can write the invariant C of p. 220 in the form a, h^ e, d, «', ft', c, «, A r> ^, «'• P/, y/> ^/» /3', y\ ^\ a •-/, .» In fact, when a member of the system \u + fiv admits of being a perfect square, we can identify \u + fiv (Art 207) with \^H + \fiH, + fi^H', and when we do so and eliminate dialyticaUy, we get this determinant. Ex. 3. In the case of Ex. 2, if Xw + fiv admit of a cube factor, X^H + \/iH, + fi^H' is the fourth power of that factor, whence by the matrix yi Yo y, a^* f. ^/, ^, -a*y = 0, can both determine the factor and the condition D = 0, Ex. 4. The qnadric covariant I comes naturally in relation with the sextio covariant 7", which may be written in any of the following forms, and which we shall now caU j =j^ +J^i^ + *c. ox* + 2bxy + cy\ ha? + 2ca?y + dy«, co* + 2rfa?y + cy« ho? + 2ca!y + ^«, tfar« + 2eiry + ey«, <&» + 2ea!y +yy« «b2 + 2<fey + cy», <^ + 2ea;y +/y«, eoc* + ^ojy + ^y« ax + by J bx + ey^ cx + dy^ — y* 6aj + cy, tfa? + rfy, <fo + cy, aiiy* cx + dyf dx + ey, ex+fy, —xhf dx + ey, eaj+yy, fx + gyj ft, ft, /, -aj2y. -y* -aj'y = i a, *, ft, c, -V, 8y2a?, -2ya:2, «», 0, ~y* 2y«aj 4a? (dbc) a? + 2 {abd) a?y + [(aic) + 3 (ocrf)] ar^y* + 2 {(ewe) + 2 (*<?(i)} je»y» + (crfe) y« + 2 (*^) ary* + [(ade) + 3 {)>ct)'\ xY, denoting by {abe) &c, the determinants as above. <h b, c », c, ^ «, (i*, e Ac. of the matrix a, d, <;, d^ e ft, <?, '^j «, / <^, <^, «, y, ^ In this notation we find / = {{abe) - 2 (crcrf)} a;^ + {{ace) - 8 (*(?<?)} ary + {{ade) - 2 (*m)} y^. But again, let the reciprocal constituents to B = Oy ft, c, d be written ft, c, rf, e Cj rf, «, f dy «, /, 9 a /, '/, Uf '/, '/ J ^/, d^ «, V, e / , 9, 280 HIGHES .3INART QUANTICS. and we have {abe)zzj\ zzg^ (aM)=8;. = -/, (We) =% = -*„ (afe) = j; = o^ Now from the identities between the constitaents of B and their xedprocali we can deriye relations among the coefficients of covariants, for instance, («;*)» = «;» - 0% + iOg*» - lOegi + 59*, -^0 = 0, {aj\ = q;; - 6*;, + I694 - 204f, + 15e/, - 6;?, + gj. = 4B. The identities got by writing the above matrix with two additional rows of its own lead to {Hj\ - ^ (if), - } (tO, + ^Al = 0. The.identities of note Art. 263 applied to the matrix in this Example aze «A-J>s ^fi,-9fi4 ffi,-9P, ' and lead to the set of relations of the type Ex. 5. The definition of I, ai^ - 4^ + 6ci, - 4dii + 6to = 22b ^ combined witk the relations at, — 8^ + Sct'i - ^t, = &c, (Art. 258), give the following identities: at, — 8W, + 8ct, — rfi, = 0, at4 — 3W, + Sctj - rf»\ = f ^ W, - Set, + 8rft, - ci, = - ii;>, «4 - 3<?t, + Sdij - et4 = jZi, ct, - Sdt, + 3«t, -/to = - */i, ct4 - Srft, + 3ci, -/t, = ^Z„ rft,-3et, + 3/t,~^=:-lii, rft4-3«t, + 3/i,-^ti:=0, which, being solved for the coefficients of t, lead to the relations Bio = f (^'2 - 2/ J, + t/o) - tW»« Ac., whence, by the last formula of Ex. 4, 75 UJi - ^JJt + 3/2^ + 65to = /,« Ac Ex. 6. If we compare the valnes of «, — 4rf, 6c, —4^, a /, -4e, 6rf, -4*, * ^» -4/» 6«, -4<i, c a, hf c, (^, 6 6, tf, d, e, / ^> d^ ^ fi 9 wofind 6 0*J.-6;J. + 15;j4-10;,«) + |(io4-^i^ = /3-i^t + A^» (Art. 257), whence 80 0*J. - 6yj, + 15yj4 - 10^,^) = 2AB-C, Ex. 7. If the three quartios in Art. 263 be the second derived functions of the aextic in Art. 262, the sextic covariant is the/ of the sextic a^ +..., and the qnadnc covailant 9 is its Z. Using the letters y and I in this sense, we have [^k/e] = ^We - 6a Jg + IbaJ^ - 20aJ, + IbaJ^ - 60^/1 + a^V fuid = 4 «0> «1> «2» «1> «2> «» Oj, a„ «4, a,, 04, «5» «, J, c, d, e a', 6', (/, d'. e Oo, < 3fi» o» «» 04 «i» < "ll «8I «4> «5 a,,. < I,, 04, as. «6 a. a. = [«o/J + i (M - 2p,Z, + M)- EXAMPLES. 281 Ex. 8. If «, V be any twolHnar7CubicsflUB*+..., a'a^+..., and if opeiating witli * fl^ "~ dxdifl "*"*" ^^ *^® product uVf we determine tlie result to be proportional to tt ; we find that v must be the evectant of the discriminant of «. This fact has already been employed to establish a canonical fonn for the sextic (Art. 174). The determinant of that article may be found, as in Ex. 1, Art. 212, by identifying the cubicoyariant 12^ of the sextic and an arbitrary cubic ax? + 8j3x^ +.,. with 'thla cubic and ^iminating its coefficients. Thus a^ - Sby + Sefi^da = Spa, bd -Sey + ddfi -ea- 8p/3, cd - 3dy + Sefi -fa = 3/oy, di-8ey-{- SJfi- ga = Spd, Now the determinant which results, <h ft, <h d+Bp A, *, d-p, e <?, rf + /», «, f -8/0, «» /, a is unaltered by changing the sign of /o, whence it is a function of p^ only, and its value is easily found = 5 + Ap^ + 9/9*. The canonical form thus amved at may be written, with the relation « -f • +io= 0, as aiifi + ^ + cafi + Bduvw {v — w) {to — u) (« — ») = 0. By conyorting this into a binary system, we find -4 = 6c + ca + o6-9<P, B zz - {be + ca + ab) d*, whence the detenninant just written breaks up into the factors {9p^ + bo + ea + ab) 0»» - (P). From the value of the oovariant it can easily be found that Z=: [abe + 2da(b--c) + d*{ba-b- e)] u* + [abo + 2i» (o - a) + (P (66 - « - a)] »» + [abe + 2(fc (a - *) + (P (5c - a - b)] to», whsnceby «C+4ilB=:4(^«2-V)=3o***<^*+2(P{6ad<?(a+*+<j)-2(62o»+cV+a«62)} + 12<i»(*~tf)(<T-o)(o-6)~9rf«{a« + J2 + o»-2(ftc + co + o*)}, WBget C=o«6V + 6aJc(o + 6 + tf)rf2 + 4(6-<?) {c-a) {a-b)d*-B (a+* + c)»<^. The binary sextic haa just been expressed by the intersections of a line with a ternary sextic. The change to expressing it by the intersections of a ternary quadrio and cubic may be made thus. Let «* = », »* = y, to* = a ; then, if 2tw = a: — y - «, 2wu=p—z — Xj luv — z — X — y, the relation «( + v + to = becomes the single relation 4^ + 4y + 4^ = ^i <^^ ^^ sextic becomes aa;» + *y2 + oas* - 3rf (y - «) (a - 0?) (aj - y) = 0. Comparing with Curves, Arts. 22(V~1, we have the values for this of the ternary invariants, S=—B and T=: C, whence the discriminant of the ternary oubic is C^—Q^B*. Ex. 9. The invariants of the system of quadrics p, I, m (Art. 262) have all been given in terms of A^ /j, /j, K with the exception of Pum^-^p^m^ +P^o and the skew invariant {plm). It is easy to show that i?o«», - 2piW, + p^m. = 90/, + 6 Jj/, - 20ItK - bA^K* + 80JP», 00 282 HIGHEB BINARY QUANTICS. whence we can expfress the square of {phn) by the others. This invariaiit {plrr^ is of the ninth degree in the coefficients of each of the quartics as in Art. 222, and the inTariant E of the sextic contains it as a factor, for, by the linear relation among p, j; •!, II, we have 6 (&•») = 72 (2/, - I^K + K') (pfoi). Ex. 10. If the second qoartic v of Art. 262 be the Hessian of the first t/, the oombinant j> ranishes identically, and the functional determinant becomes the sextic ooraziant of the biquadratic «. The identities of that article show that the coTariant t of this sextic vanishes identically {CUbschy p. 447). Ex. 11. If the two quartics u and v of Art. 262 be the derived functions of a quintic, the Jaoobian is its Hessian, and their invariant B (Art 220) vanishes identically. Now in the present notation we have from that article A +48£ = - 40^4,, A — 12^ r:-4P ; thus for £ = 0, P = 10i4f : whence this invariant relation among the quartics is the same as that K= — ^A^ shall satisfy the quintic of Art. 262. (Stephanos, Lc. p. 81 ; F. Lindemann, Math. Ann, xxL, p. 81.) Ex. 12. If a quintio and a sextic admit of reduction to the forms A'^ + Bv^ + C'ufi ■{- lyz^, Atfi + Bifi + Cvfi -{■ Dsf^y they aatisfy the invariant relation «', »'. o. i, e »', <f, «, «. d e*. f, «> d; « *. «', d, •> ,/ = 0. Ex. 18. In order to reduce two quintics to the forms Av^ + Bv^+Cv^-{- Dafi, A*v^ + B^t^ + C"w» + Z/^*, their canonizant uvun is = 0. ax + by^ hx + ey, a'x-k-b'y, Ifx + c'y bx + eyf cx + dy, b'x + c'y, c'x + d'y cx + dyj dx + eyf <fx + d'y, d'x + e'y dx + ey^ ex +fyf d'x + e'y, e'x -\-f'y j Ex. 14. Similarly if a cubic and quintic admit of reduction to the sum of tbi^ cubes and three fifth powers of the same quantities, we have a\ a, b, c \— 0. Cy d d, e cf , rf, «, / Ex. 15. In Hermite*s invariant, p. 234, the leading terms in a and/ are / = ay (rf/- e«)» ^af^{ac- b^^ + «kc. Ex. 16. Determine a quintic v such that the result of operating with 12^ on uvt where u is a given quintic, may vanish identically. We find *', ft, C, c, ^ -8c, 3b, -a. «» ^2d, 0, 26, -o. /, «» -8J, 8(?, -ft, — a 0, / -2e, 0, 2c, -b 0, 0, /, -3e, Sd, — c which is the product of u by its quartinvariant = — uJ of p. 228. ( 283 ) ..., LESSON XIX. ON THE OaDER OF BBSTBICTED STSTEMS OF EQUATIONS. 267. The problems discassed in this lesson are purely alge^ braical, and in the investigation of them I do not make use of any geometrical principles. But I find it convenient to borrow one or two terms from geometry, because we can thus avoid circumlocution, and also can more readily see how to extend to quantics in general theorems already known for ternary and quaternary quantics. We saw (Art. 78)* that if we are given k equations in k independent variables, the number of systems of common values of the variables which can be found to satisfy all the equationS| will be equal to the product ot the orders of the equations. Now, in the geometry of two and three dimensions respectively, the system of values a? = a, f/^b] ora? = a, y = i, « = c denotes a point. I find it convenient therefore to use the word " point " in general instead of ^^ system of values of the variables," so that the theorem already stated may be enunciated : ^^ A system of k equations in k variables of degrees Z, m^ n, p, j, &c respec- tively, represents Imnpq &c. points^'* by which we mean that so many ^^ systems of values of the variables " can be found to satisfy all the equations. This number Imnpq &c. will be called the order of the system of equations. 268. If we have a system of i — 1 equations in k indepen- dent variables, we have not data enough to determine any system of common values of the variables, and the system of equations denotes a singly infinite series of '^ points." Such a system of equations we shall speak of as denoting a curve* If with the , given system of A — 1 equations we combine any arbitrary * If as is usual we employ homogeneous equations, the number of variables wQX of course be A: + 1. 381 ORDSB OF RE8TBICTED SYSTEMS OF EQUATIONS. equation of the first degree, we have then data enough to determine points which will be equal in number to the product of the degrees of the equations. We shall define the order of a curve as the number of points which are obtained when^ with the equations which denote the curve, we combine an arbitrary equation of the first degree. When we are given a system of A; — 2 equations, .these denote a doubly infinite series of points, since we cannot de- termine any points unless we are given two other equations. Such a system we shall speak of as denoting a surface. If with the system of i!; — 2 equations we combine an arbitrary ^[uation of the first degree, we shall have a ^^ curve " whose order is the product of the degrees of the k—2 equations. In general, by the order of a surface^ we mean either the order of the curve obtained by combining with the given equations an equation of the first degree, or, what comes to the same thing, the number of points obtained by combining with the given equations two equations of the first degree. And so, more generally, if we have any system of fewer than Jc equations, by the order of the syst^em we mean the number of points that are obtained, when with the given equations we combine as many equations of the first degree as are wanting to make the entire number of equations up to A;, thus affording data enough to determine systems of values of the variables. It is evident that in the case under consideration, the order of the system is the product of the degrees of the equations which compose it. 269. If we have A; 4- 1 equations in h independent variables whose degrees are 2, tti, n, &c., we can eliminate the variables; and we have seen (Arts. 76, 78) that the order in which the coefficients of each equation enter into the resultant, will be equal to the product of the degrees of the remaining equations Taking then, to fix the ideas, the case of four equations: let their orders be Z, 9n, n, r, and let any quantity enter into the coefficients of the equations in the degrees X, /l6, v, p respec- tively, this quantity will enter into the resultant in the degree \mnr + finrl + vrlm + plmn. EXPLANATION OP TERMS 285 We shall use the word order to denote the degrees Z, m, w, r, in which the equations contain the variables which are to be eliminated, and loeight to denote the degrees X, /l6, v, p in which they contain the quantity not eliminated ; and the result just written may be stated, that the weight of the resultant, or the weight of the system, is equal to the sum of the weights of each equation multiplied by the order of the system formed by the remaining equations. And this is still true, if we break the given system up into partial systems. Thus, the first two equations form a system whose order is Im and weight Xm + fil^ and the second two equations a system whose order is nr and weight it + pn ; and the value just given for the weight of the entire system is nr (Km + fil) + Zm (vr + pn)j that is, it is the sum of the weights of each component system multiplied by the order of the other. The advantage of so stating the matter will appear presently. 270, What has been hitherto said in this lesson is but a re-statement in other words of principles abeady laid down in the lesson on Elimination ; but my purpose has been to make more intelligible the object of investigations, on which we shall now enter, as to the order and weight of systems of a somewhat different kind. We have seen that k equations in k variables represent Imnp &c. points. But now we may combine with these k equations an additional equation, which is satisfied for some of the points but not for others of them. We have then a system of &+ 1 equations representing points, that is to say, all satisfied by a number of systems of common values of the variables, that number being now, however, generally smaller than the product of the degrees of any k of the equations. Cases are of constant occurrence where a number of points can be expressed in no other way than that here described. A simple geometrical example will suffice. Consider p points in a plane where ^ is a prime number, and where the points do not lie in a right line, then these points cannot be represented as the com- plete intersection of any two curves, and if we have any two curves going through the points, their intersection includes 286 ORDER OF RESTRICTED SYSTEMS OF EQUATIONS. not only these points but others besides. To define the points completely, we must add a third curve going through the p given points, but not through the remaining points of intersection of the first two curves. The points are thus completely defined as the only points common to all three curves. Our object then is, in some important cases where a system of points is defined by more than h equations, to lay down rules for ascertaining the order of the system; that b to say, how many systems of common values satisfy all the equations. In like manner a system of A; — 1 equations is satisfied by an infinity of common values. But it may happen that we can write down an additional equation satisfied by part of this series of common values, but not by the remaining part. In such a case, the system of A; — 1 equations denotes a complex curve, and it requires the system of Tc equations to define that part of it for which all the equations are satisfied. It will be the object of this lesson to ascertain the order and weight of what we may call i*estricted systems; that is to say, where to a number of equations sufficient to define points, curves, &c., is added one or more others which exclude from consideration those values of the variables which satisfy the first set of equations, but do not satisfy the additional equations. 271. The simplest example of such a system is the set of determinants Uj Vj w =0, u\ v'j w' or, at full length, vw' — wv' = 0, wu' - uw' = 0, uv' — vu' = 0. By writing these equations in the form u V to vi v' w' ' it is evident that, in general, values of the variables which satisfy two of the equations must satisfy the third. But there is an exception for the case of values which make either u and u'l t; and v\ or w and to => 0. In any of these cases it is easy to see SYSTEM OF A VANISHING MATRIX. > 287 that two of the equations will be satisfied, but not the third. And BOW it is easy to see how to calculate the order of the system common to all three. Let the orders of u and v!y of v and v\ of w and w\ be Z, wi, n respectively ; then the orders of the first two equations are w + w, w + Z, and of the system formed by them is (m + 7i)(n + 7)* But in this system will be included values which satisfy both w and w\ these values not satisfying the third equation. Excluding then this system, the order of which is n', the order of the system common to the three determinants is mn + wZ + Im. In like manner, suppose we have a system with three rows and four columns, u, u\ w", It'" = 0. r, v', t/', v'" Let us write at full length the determinants formed by the omission of the third and fourth columns u" [vw' - wv') + v" {wu' - uw') + M?" (wt?' - vu') = 0, u''' [vvf - vw') + v"' [wu' - mo') + 1^'" [uv' -vu") =0, then these two equations are obviously satisfied for all values which satisfy the three vw =^wv\ wu' ^uw\ uv'=^vu\ But these values will not satisfy the other determinants of the given system. From {l + m-\- nf, then, which is the order of the system formed by the two equations written at length, we must subtract tw/i + wZ+Ztw, which has just been found to be the order of the system special to these two equations, and the remainder P + w' + n* + win + nZ + Zw is the order of the system common to all the determinants. Having thus determined the order of a system with three rows and four columns, we can, in like manner, thence derive the order of a system with four rows and five columns. Proceeding thus step by step we arrive by induction at a general formula, for the order of a system with A; rows and (A + 1) columns. 272. We may consider in succession the cases : 1**, k roT^s and Jc columns; 2'', k rows and [k+l) columns; S""^ k rows and A; + 2 columns, and so on. Writing down in each case only the. 288 ORDER OF RESTRICTED SYSTEMS OF EQUATIONS. I a + a\ =0, = 0, &c., II a + ex, b + a ||=0, = 0, &c., ||a+a,&+a,c+a||=0, = 0, &c., orders of the several ftmctions, so that a + a^b + /S, &g.^ stand for fiiDctioQs of the orders a-^a^ b + ^j &c. respectively ; the case 1* indudes the systems a + a, b + a a + fi, b + 13 the case 2"" indades the systems a + a, & + a, c + a a + /3, b + 13, c + 13 the case 3* Includes the systems a + OLj b+a, c + a, d+a a+iS, J+iS, c+^, d+fi and 80 on. Write in each case so that Cj, (7„ C7, ... denote the sums of the products without repetitions of the letters a,b, ... (as many of them as belong to the system in question) taken one together, two together, three together, &c, and 5; = 2a, S, = 2a' + 2ai8, S, = 2a' + 2a'/8 + Sa/37, •.., or let H^jh^^Hy., denote the sums of the homogeneous products, with repetitions of the letters a, /3, ... (as many of them as belong to the system in question) taken one together, two together, three together, &c. Then in the case 1", the order of the system Is = (7^ + jBJ, 2°, „ „ = (7. + C7.fi; + jg;, and so on. Thus in the case 1°, there is only a single equation ; and for the several systems written down above, the orders are a + a; a + b + a + ^j &c., thus for each system the order is G^ + H^. In the case 2", for the first of the systems written down above, there are two equations of the orders a + a, b + a re- spectively, and the order of the system Is = (a + a) (6 + a), and this is = aJ + (a + J) a + a*, which i8 = (7,+ G^H^ + E^. CASES OF k BOWS AND it + 1 COLUMNS. ^9 For the secoDd Bystem, viz. the system ll a + a, 6 + 0, c + a l{ = 0, II a + /9, J + i8, c + fi II vrhieh incladeB the first of those coDsidered in last article, applying to it the reasoning of that artide, we have two equations of the orders a + J + a+/S, a + c + a + /9 respectively ; the prodnct of these numbers is = <^ + a[b-t-c) + bc + (2a + b + c) (o + ^8) + (ot' + 2ajS + (S*) ; but we have to subtract from this the product (a -Ha) (a + j3), which is = a* -I- a (a + ^} + 0/3 ; and the order of the system is thus found to be=ai + ac + ftc + (a + 6 + c)(a + /3)+a' + ^ + oj8; which is = a, + C7, fl, + a;. The next system is I a + a, J + a, c + «, d + a. a + ^, 6 + /9, c + j3, d + I a + y, h + y, c + y, d + y the order of this is equal to the order of the systt a+flt, i + o, c + a = a + ff, ft + /9, c + ff « + y, J + 7, + 7 leas the order of the system II a + a, yS + a, 7 + a || = 0, II a + i, j8 + i, 7 + 6 l| TIi^ and this is = {a + 5+c + a + /? + 7)(a + 5+d+a + j3 + 7) _{a' + J' + oJ+((i + J)(a + /3 + 7} + (o/3+a7 + j37)} - (a + J+c)[o + 6+rf)+(2a + 2j+c+(iKa+;3 + 7}+(a+/S+7)* -a*-6'-aJ- (a + 6) (0 + ^8 + 7) -0^8-07-^87 = ab + ae + ad+be + bd+cd+{a + b + c + d){a + ^ + y) + a' + /3'+ 7'+ 0/3 + 07 + ^97, which is = a, + C.ff, + S„ and similarly for the other systems of the case 2*. 290 ORDER OF RESTRICTED SYSTEMS OF EQUATIONS. Id the case 3"*, for the first system we have three equations of the orders a + o, & + a, c + a respectively ; and the order of the system is {a + a) (& + a)(c+a), =aJc + (aftH-ac+ic)a+(a + i + c)a' + a* and the result may be verified for the other systems. 273. We may proceed in like manner to calculate the weight of the system of determinants considered in the last article. Beginning again with the simplest case, let us suppose that the system u, t;, to u\ t?', w' is to be combined with one or more other equations and the variables eliminated. Now the result of elimination between uv' — t?u', uv/ — wu'j and any other equations will contain as a factor the resultant of Uy u\ and the other equations. If we reject this factor we get the same result as if we had eliminated between uv^ — vu\ vuf — wxf and the other equations, and then rejected the factor got by elimi- nating between t;, t;^, and the other equations. To illustrate the method employed, let us suppose that u, ii\ t?, vf \ Wj va respectively contain any quantity not eliminated in the degrees X, /L», 1^; and that we are to combine with the determinants of the given system another equatioa jS = 0, whose order is r, and containing the uneliminated quantity in the degree p. This quantity then will enter into the resultant of R, mv'— tn*', uuf- wv! in the degree p(Z + w)(Z+7i)+r{(Z + w) (X + F) + (? + n)(X + /it)}. But the resultant of R^ u, u', will contain the same quantity in the degree pP + 2rlk. When, then, this factor is rejected from the former result, the remainder is p [mn + nZ + Zwi) + r {X(m + w) + /i (w + Z) + F (l-\-m)\ The order then of the system of three determinants is the quantity multiplying p, and the weight is the quantity multi- plying r. CONDITIONS FOB TWO COMMON ROOTS* 291 274. Finding in this way the weight of any system of those considered in Art. 272, the result is that if the orders of the several functions be as written in Art. 272, and if their weights (that is to say, the degrees in which they contain the vari- able not eliminated) be a' + a', 6'+ a', (fee, a' + iS', &c., then the formula for the weight is derived from that for the order, by performing on it the operation 275. If we form the condition that the two equations a<' + Jr' + cr' + &c. = 0,aT + 6'^"*"' + c'^'^ + &c. = 0, should have a common root, we obtain a single equation, namely the resultant of the equations. But if we form the conditions that they should have two common roots, we obtain (Art. 82) not two equations, but a whole system, no doubt equivalent to two conditions, yet such that two equations of the system would not precisely define the conditions in question. Now we may suppose that ^ is a parameter eliminated, and that a, by &c. contain variables, and we may propose to investigate the order of the system of conditions in question. Now, Art. 82| these conditions are the determinants of the system a, &, c, dy ... O, Oy c, • . . a, by « • • a, by C, d y ... /»' 7/ g%' 9 9 9 *** where the first line is repeated m - I times and the second Z - 1 times; there are ?+»i--2 rows and ?+w-l columns. The problem is then a particular case of that of Art. 272. We suppose the degrees of the functions introduced to be equi- different; that is to say, if the degrees of a, d be X, /li, we . suppose those of i, J' to be X + a, ft + a ; of c, c' to be X + 2a, , /i + 2a, &c. The formula of Art. 272 is, order =*(?,+ GJd^ + if,. 292 ORD£R OF KESTRICTED gTSTEMB OF EQII^TI0V& To apply it to the present cftse we may take for the quantitiai Qj i, Cj &c. of Art. 272, 0, a, 2a, &c. ; and for the quantities a, fij 7, &e., X, X — oc, X — 2a, &c. C^ is then the sam of Z+m-2 terms of the series a, 2a, 3a, &c., and is therefbrs if we write I-^m^kj ^(^ — 1)(A; — 2)a. In the same eass C7, is the sum of products in pairs of these quantities, and is therefore (^,8)(*-2)(A;>l)(3*-4) 1.2.3.4. Again H^ is the sum of m — 1 terms of the series X^ X — a, X - 2a, &c., and of 2 — 1 terms of the series ^, /* — a, /* — 2a, &c We have then ir, = (m«l)X + (Z-l)M-ia{(w-l)(t«-2) + (/-l)(f-2)}. Moreorer B^^^ [H^ + 5,), where 8^^ the sum of the squares of the same m — 1 and I- \ terms is «(»i-l)X»+'(Z-l)/i'-^a(in-l)(«t-2)-/*a(/-l)(i-2) [ (m^l)(7n^2)(27ii^3) (f- 1) (f-2) (2f--3) ] ( 1.2.3 1.2.3 r Collecting all the terms, the order of the required system is found to be \m (m - 1) X" + \l (I - 1) / + (Z - 1) (m - 1) X/a + im (»i - 1) (2? - 1) Xa + i^ [l - 1) (2W - 1) /la + iZw(Z-l){«w-l)a'. If the two equations considered are of the same deg^e, that is to say, if Z = «i, we may write X+/iA=p, y^fJ^^q^ and the order becomes \m [m — 1) (^ + wia) {^ + (»i — 1) a} — (wi — 1) q. If all the functions a, J, &c. are of the first degree, writing X = /A = 1, and a = in the preceding formula, the order is found to be i (Z + ?n - 1) (Z + m - 2). 276. If the ilegrees in which the uneliminated variables occur in any terms be denoted by the accented letters corresponding to those which express their degrees in the variables to be eliminated, then the formula for the weight of the system is FACTOR COMMON TO THREE EQUATIONS, ?93 obtained from that for the order by performing on it the. opera- tion ^'jT + /*';j~ ■♦• ^';r • I^^ other words the weight is wi (m- 1) XX' + Z{Z- 1) /i/A' + (Z- 1) (m- 1) (X/a' + /aX') + iw (m - 1) (2Z- 1) (Xa'-f aX') + i^C?- 1) (2m - 1) (/la' + a/) + ?«i(Z-l)(7n- l)aa'. 277. The next system we discuss is that formed by the system of conditions that the three equations ai + Jr' + &c. = 0, a r + J't"*-' + &c. = 0, a"«* + Vr^ + &c. = 0, may have a common factor. The system may be expressed by the three equations obtained by eliminating i in turn between every pair of these equations, a system equivalent to two con- ditions. Thie order of the system may be found by eliminating from the equations the variables which enter implicitly into a, 5, c, &c., when the order of the resultiug equation in i deter- mines the order of the system. Let us suppose that a, a\ a'' are homogeneous functions In a;, y, z of the degrees X, /*, v respectively ; that ft, h\ i" are of the degrees X — 1, /:* — 1, v - 1, &c., and if we take the reciprocal of < as a fourth variable, the equations are of the orders respectively X, /i, v, forming a system of the order X/av. But the system of values a; = 0, y = 0, a = is a multiple point in the three equations of the orders X-Z, /a— w, v — n respectively. The order then is to be reduced by (X — Z) (/t — m) (v - n). It is therefore lliv + mvK + wX/A — \mn - /tnZ — vim + Imn. This then is the order of the system we are investigating. If the orders of i, h\ c, c', &c. had been X + a, /^ + a, X + 2a, Ik + 2a, &c., then the order of the system would have been lykV + wivX + nX/A + a («inX + nliu + Imv) + aflmn. The weight is found by operating on this with ^ -rz^ &C') ^Q^ is I (/a/ + vii!) + m (vX' + Xv') + 71 (X/ H- /aX') + mn (aX' + Xa') + wZ (a/A^ + /na') + Zwi (ay' + va') + 2Zinnaa!!* 294 ORDER OF RESTRICTED STSTEM8 OF EQUATIONS. 278. It is a particalar case of the preceding to find the order and w^gfat of the system of conditions that an equation ol^+ (^+&c = may have three eqoal roots; because these oondidons are fbond by expressing that the three second differen- tials may have a common fiustor. Writing in the preceding for /, m ; for fi, n - 2 ; for /ft, X + a ; and for v, X + 2a ; we find, for the order of the system, 3(H-2)X(X + na) + n(H-l)(n-2)a'; and In like manner for its weight 6 (n-2)XX' + 3ii (n- 2) (Xa' + aX') + 2ii(n-l) (n - 2)aa'. Again, to find the order and weight of the system of condi- tions that the same equation may have two distinct pairs of equal roots, we form first, by Art 272, the order and weight of the system of conditions that the two first differentials af^'^&c.f 6^ + &c. may have two common factors. We subtract then the order and weight of the system found in the first part of this article. The result is that the order is 2(n-2)(ji-3)X(X+na) + i»(n-l)(n-2)(n-3)a*, and the weight is 4 (h-2) (n- 3) XX' + 2n (n -2) (n - 3) (Xa' + aX') + n (n - 1) (n - 2) (n - 3) out. Before proceeding further in investigating the order of other systems, it is necessary to discuss a different problem, and I com- mence by explaining the use of one or two other terms which I borrow fix)m geometry. 279. Intersection of quantics having common curves. Two systems of quantics are said to intersect if they have one or more '^ points" common, that is to say, if they are both capable of being satisfied by the same system of values of the variables. A " surface " is said to contain a " curve " if every system of values which satisfies the k—1 equations constituting the curve, : satisfies also the A; — 2 equations constituting the surface. Thus, in the case of four variables, three equations 17= 0, F= 0, TF=0 constitute a curve, and the two equations U=0 F= con- stitute a surface which evidently contains that curve. QUANTIC8 HAVING COMMON CURVES. 295 ' Now a system of k quantics in ;i variables, in general, as we have seen, intersect in a definite number of points, that number being the product of the orders of the quantics. But it may happen that they may have an infinity of points common, these points forming a " curve " in the sense in which we have already defined that word. Besides that curve they will have ordinarily a finite number of points common, which it is our object now to determine. Let us take, for example, to fix the ideas, the case of four independent variables; and suppose that we have four equations of the form U = Au +Bv -^ Cw =0, V=A'u +B'v +C'w =0, Pr= A''u + B''v + C'w = 0, Z = A'^'u + B'^'v + C'^w = 0. We suppose the degrees of ?7, F, TF, Z to be Z, m, n^p ; of m, v, to to be X, /A, v; and -4, B] A\ R^ &c. are therefore functions of the degrees Z — X, Z — /i; w — X, m — ix^ &c. Now, evidently, these equations will be all satisfied by every system of values which make t« = 0, t; = 0, t(; = 0; and these equations not being sufficient to determine ^' points," will be satisfied by an infinity of values of variables. In other words, the four quantics [7, F, W^ Z have a common curve uvw. And yet ?7, F, PF, Z may be satisfied by a number of values which do not make t^, t;, w all = 0. It is our object to determine this latter number; and our problem is, When a system of quantics has a common curve, to find how many of their Imnp^ &c. points of intersection are absorbed by that curve, and in how many points they intersect not on that curve. 280. Let us. first consider the curve formed by A;— 1 of the quantics; for instance, in the example we have chosen for illustration, the curve Z7FIF. Now evidently a portion of this curve is the curve uvw^ but there are besides an infinity of points satisfying UVW which do not satisfy m, v, w. We speak then of the curve J7FTF, as a complex curve consisting of the curve wi?i^ and a complementary curve. Now the order of a complex curve is always equal to the sum of the orders of its 296 ORDER OF RESTRICTED SYSTEMS OF EQUATIONS. oomponents. For, by definition, the order of the complex curve UVW 18 the namber of points obtained by combining with the equations of the system an additional one of the fiirst degree: that order being in the present case Imn. And eridently, since of those Imn points Xf^y lie on the carve uviVj there most be Imn — Xf^v on the complementary carve. The two carves intersect in points whose number t is easily obtained. For evidently all points which satisfy the three equations and which do not satisfy UjVjW] must satisfy the determinant A, B, C A\ B\ C A'\ B'\ O" = 0, the degree of which isZ + m-fn- X— /4 — f. The intersection of this new quantic with umo gives all the points in whidi wow meets the complementary curve. We have therefore t = \yi,v (Z + w + n — X — /Lt — v). 281. To find now the number of points common to UVWZ^ we have to consider the points in which the curve UVW meets Z; and it is required to find bow many of these are not on the curve uvw. But since uvw is itself a part of the curve UVW^ it is evident that the points required are contained among the p [Imn — \fiv) points in which the complementary curve meets Z. And from these points must be excluded the f points in which the complementary curve meets uvw. Using then the value given in the last article for t, we find, for the number which we seek to determine, Imnp — Xfiv (?+ m + w +p) + Xfiv (X + /a + v). We shall state this result thus, that if k quantics of orders Ij 7n, n, Pj &c. have common a curve of order a; then the number of points which they will have common in addition to this curve is less than the product of the orders of the quantics by a (Z + m + n + &c.) — fij where )8 is a constant depending only on the nature of the curve and not involving the orders RANK OP CURVE^ 29^ of the qualities. We shall call this donstant the 7*ank\)t the cairve. We have seen that when the curve is given as th^ intersection of quantics u, v^ w^ the order is \/liv and the i^nk X/iV (\ + /Ll + v). We saw, in the last article, that if the intersection WW consists of two complementary curves whose orders are a, a\ and whose ranks are iS, ^^ the number of points in which the two curves intersect is a (Z -f «i + n) — ^ ; and by parity of reasoning it is a[ {l-^m^-n)— p^] Hence the orders and ranks of the two complementary curves are connected by the equa-^ tions a + a' = Zmn, ^ — ^8' = (a — a')(? -h w + n). 282. Next, let us consider the case where the qnantics have common two or more distinct curves uvw^ u'rfw\ &c. Let the intersection for instance of WW consist of the two curves uvtD^ uvw\ and of a complementary curve d' ; then, in the first place, the order of a" is evidently Imn — X/iaf — X'/aV- Secondly^ we have seen that wow meets the remaining intersection of WW in points whose number is X/iv (Z -H wi + w — X - /4 — v). If then i of these lie on u'x>vf (that is to say, if wow^ ttVt^ intersect in t points) there must be on the complementary curve (^' \fAV (Z + Wl + W — X — /lA — v) — t. And in like manner a!' meets wVi/?' in X'/iaV (Z + »» + n — X'— /a' — v') — i points. As before, then, the number of points on neither curve in which oC^ meets any other quahtic Z is (Z/wn — X|iv — X'/aV) p — X/Ltv (Z + m + n — X — /ia — v) - WikV{l-^m + n - X'- /- •) + 2*; or Imnp — (X/[av + X'/[aV) (Z + m + n -{-p) + X/av (X -f /h + f) + X>V(X' + /ia' + /) + 2i. Thus, then, the diminution from the number Imnp effected by a complex curve is equal to the sum of the diminutions effected by the simple curves less double the number of their points of intersection* The same holds no matter how many t896 ORDEB OF RBSTfilCTED SYSTEMS OF EQUATIONS. be the carves common to the quantics; and we mav say that when a complex curve consists of several simple curves the order of the complex is eqaal to the sum of the orders of its components ; and the rank of the complex is equal to the sum of their ranks increased by doable the number of points common to every pair of curves. 283. We give, as an illustration of the application of these principles, the problem to determine how many surfaces of the second degree can be described through five points to touch four planes. Let 8^ Tj Uj V^ W he five surfaces passing through the five points, then any other will be of the form aS+fiT-^^yU+SV+^W] and the condition that this should touch a plane will be a cubic function of the five quantities a, i8, 7, 5, e. We are given four such equations, and it is required to find how many systems of values can be got to satisfy them all. If the four equations had no common ^^ curves" the number of their common "points " would be 3* or 81. But the existence of common curves may be seen in this way : The condition that a surface of the second order should touch a plane vanishes identically when the surface consists of two planes. Let us take then for 8 and T two pairs of planes passing through the five given points, 5= (123) (145), r= (123) (245); then evidently, the condition that a8+ f3T-\-yU'\- SV-i- eW should touch any plane whatever, must be satisfied by the sup- position 7 = 0, 8 = 0, 8 = 0. This " curve," then, which is of the first degree, will be common to all four quantics. And, if we call this the line (123) (45), it is evident, by parity of reasoning, that the quantics have common ten such lines (124) (35), &c. Now if, as before, we take 8 as the system of two planes (123) (145), r= (123) (245), and take U= (145) (234) ; then, while the line (123) (45) is denoted by 7 = 0, S = 0, s = 0, the line (145) (23) is denoted by /8 = 0, S = 0, s = ; and these two lines intersect, being both satisfied by the common values /g = 0, 7 = 0, 8 = 0, 8 = 0. And, in like manner, (123) (45) is intersected by (245) (13), (345) (12). Thus, then, the ten lines have fifteen points of mutual intersection. The rank of a single curve of the first degree being got by making \ = f( = v = lin •GEOMETRICAL ILLUSTRATION FROM QUADRICS. 899 the formula \fiv (\ -h /c^ + v) is three. Hence the rank of the entire system is ten times three increased by twice fifteen or is 60. And the number of points which satisfy the four quanticB is 81 - 10 (3 + 3 + 3 + 3) + 60 or is 21. 284. We have shown, Art. 272, how to determine the order of a system of determinants, the number of rows and columns in whose matrix differ by one. We shall now show how, in the last mentioned case, to determine the rank of the curve. Com- mencei as before, with the simple case tf, v^ w w', v', w' and we see that the intersection of uv' — vvk^ uuf — wvl is a complex curve, consisting of the curve uvl and of the curve with which we are concerned, and knowing the order and rank of uv!^ we find the order and rank of the other curve. Repre- senting as before the orders of the several terms by a + a, J + a, c + a a + /3, 6 + i8, C + /9 we thus obtain lUnk = rank of (wt/ — ijm', uvi — wit') — rank of (u, u') —twice number of intersections of the two curves^ and this is = (a + 6 + a + iS) (a + c + a + /9) (2a + 6 + c + 2a + 2)8) -(a + a)(a + /3)(2a + a + /3) - 2 (a + a) (a + iS) (i + c + a + /3), or, introducing the former notation (see Art. 272), Cj = a -f 6 + c, &c., 5"^ = a + iS, &c., thisis = (a + 6+S;)(a + c-f J?J(a + (7j + 25J. -(a' + afi; + 5.*-^J(2a,+ 3fi;)j or, what is the same thing, = {a'+C; + (a + aj5;H-fi/}(a+ai + 2fi;) - (a* + aE, + fl.' - 5-.) (2 (7, + 3S;), 806 OSDEB OF RESTSICTED STST£MB OF EQUATIONS, whidi is eanlj found to be or attending to the relation H^ — ^^i^t + B* = which exists in the case of two equations (a, ^8), this is + c, (^/ + 25;) or, finally, the rank U = (C7.+ Cfi,+ C,H, + H,) + (0,+ C,E, + H;) {C, + ff,), and passing successivelj to the cases of four columns and three rows, fiye columns and four rows, &c., it maj be shown that C7,| C7,, C7g referring to the series of numbers, a, &, c, &c., and ff^j H^j H^ to the series of numbers, oc, /3, &c., the foregoing expression for the rank holds good for the system in which the number of the rows and columns differ bj one. 285. The formula of the last article may be applied to calculate the order of the system of conditions, that the equa- tions af*^ + &c., df + &c. may have three common roots. The conditions are formed by a system of determinants, the matrix for which is formed as in Art. 275 ; save that the line a^ bj c is repeated n — 2 times, and the line a', b'j c', m — 2 times. The matrix consists of w + n - 2 columns and w + n — 4 rows. The order of the system then calculated by the last article is found to be 7i(n-l)(n— 2) ^.. rwfw— l)(aw — 2) . ., ^», ^., ^.^, 1.2.3 ^ '^ 1.2.3 ■>"+M^-l)(«-2)(w-2)XV 4 i (w- 1) (m-2) (71-2) X/AM-i(w - 1) n (n- 1) (n- 2) X'a + i(n-l)7n(7n-l)(7n--2)/i'a+ j^[m-2){n-2) {m (n-l)+n («i-l)}X/ia + {^^ (w - 1) (w - 2) tn (w - 2) ■\-in{n'- 1) (n - 2)} a*X + {iin(m- l)(»i-2)n(n-2) + in(m-l)(7n-2)}aV + im (w - 1) (m - 2) w (n - 1) [n - 2) a'. jTWO JSQUATIONS .WITH THREE ROOTS COMMON. S0\ In the case where we have a = 0, X = /i* = 1, this reduces to i(m + n-2)(m + n-3)(m + n-4). The weight of the sjstem| found by the same process as before is in(n-l)(n-2)XV + im(iii-l)(«i-2)/iAV + (w-l)(n-2)(m-2)(X/iX'+JXV)H-(m-l)(m-2)(n.2)(X^/+J/iV) + (m - 1)« (n- 1) (n- 2) (XX'a+ JXV) + (n - 1) jn (w - 1 ) (m - 2) {fifk'a + i/t*aO + i(w-2)(n-2){2»in-m-n}(X/a + X'/Aa + Va') + {|n (n- 1) (w- 2) m (w- 2) + Jw (n- 1) (w- 2)} (aV + 2aa'X) + {^m {m - 1) (w - 2) n (w - 2) + Jm (m - 1) (w - 2)} (aV+2aa» + Jw(«i-l)(w-2)n(n-l) (n-2)aV. 286. The next problem we investigate is when a system of quantics have a " surface " common, to find how many of their points of intersection are absorbed by the common surface. We mean by the order and rank of a surface^ the order and rank of the curve which is the section of the surface by any quantic of the first degree. Thus, consider the case of five independent variables, then a system of three equations con-> stitutes a surfiE^ce, and if their orders be X, /i, v, the order of the surface will be X/av, and its rank X/iv (X + /tt + v) ; these being the order and rank of the curve got by uniting with the given equations an additional one of the first degree. Now, first let k-X quantics have a surface in commoUi whose order and rank are a, /3; they will also in general have common besides a complementary curve whose order is readily found. Thus if A; = 5, joining with the given quantics another of the first degree, we then have a system of 5 quantics, having a curve common, and therefore by Art. 281 intersecting in Imnr -a(Z + ?n + w + r) + )8 points besides. But these are the points in which the quantic of the first degree meets the complementary curve, and therefore this is the order of that curve. 302 ORDER OF RESTRICTED SYSTEMS OF EQUATIONS. 287. Next let us investigate the namber of points in wMch the sarface and complementarj curve intersect each other. Let Z7, F, W^ Y (being as above of the orders /, m, n, r re- spectively) be respectively of the forms Au -{-Bv -^Cw ^% A'u ^-Bv +G'w =0, ^"u+5"r +(?"«? =0, ^'''u + 5"'t; +(?"'«? = 0, where w, r, to are of the orders X, /a, y respectively. Then the points common to Uj Vj Wj Y which do not make u^v^w = 0| will satisfy the system of determinants A, A', A', A B, B, B\ B' C, C, 0", C' =0. But since A is of the order Z— X, 5 of the order l—fi^A^ of the order m — X, &c., it follows (Art. 272} that the order of the set of determinants is {Im + Zn + Zr + mn + mr + nr) - ( Z + m + n + r ) ( X + /[* + v) + (X*+/A* + y* + X/* + Xv + /Av). If now we combine this system of determinants (equivalent to two conditions), with the A; — 3 conditions which constitute tbe surface, we determine the points common to the surface and complementary curve. And their number is the order of the system of determinants, multiplied by \fiv. Writing then a and fi for the order and rank of the surface X/av, \fiv (X + /* + f), and denoting by 7 the new characteristic X/iV (X* + /A* + V* + X/A + /AV + vX), which we may call the class of the surface, we find t = a {Im + Zn + ?r + mn + mr + nr) -/9(Z+wi + w+r) 288. If then we have an additional quantic Z also con- taining the given surface, and if it be required to find how naany points not on the surface are common to all 5 quantic8| QUANTICS WITH COMMON SURFACE AND CUBVE. 30S these will be evidently the points of intersection of the com- plementary curve with Z^ less the number of points of intersec- tion of the complementary curve with the surface. If then Z, m, n, r, a be the orders of the quantics, the number sought will be got by subtracting from 8 [Imnr — a (Z+ m + n + r) + /9}, the number a [Im + ?w + mn + ?r + wtr + nr) - /9 (Z-f 7n + n + r) + 7. And the di£ference is lmnr$ — a(Zm+...+r«) + 0{l+m + n + r + 5) — 7, which is the formula required. 289. Next let us consider the case {k = 5] where a system of quantics have common not only a surface, whose characteristics are a, )8, 7, but also a curve, whose characteristics are a', ^8', intersecting the surface in i points. As before, consider first 4 of the quantics. Their intersection we have seen consists of the surface and of a complementary curve, whose order is Imnr — a (Z + w + n + r) + )8. And if the complementary curve be itself complex, . consisting in part of the curve a\ and also of another curve, whose order is a", we have evidently a" = Imnr — a ( Z + wi + n + r) + /9 — a'. The points therefore which we desire to determine are got by subtracting from the aaf^ points of intersection of the curve a" with the remaining one of the given system of 5 quantics, 5 + S' where 8 is the number of points in which the curve a'' meets the surface (a, )8, 7), and S' is the number of points where it meets the curve. But we know S, since we know, by Art. 287, the number of points where the surface is met by the entire curve complementary to it ; and therefore have S + 1 = a (Ztw -h ?n + &c.) -i8(? + w+w + ^)+7; and We know 8', knowing, by Art. 280, the number of point* 804 OtCDER OF RESTRICTED SYSTEMS OF EQUATIONS. in which the curve a' is met by the entire curve complemeiitflrj to it, and therefore have S' + 2i = a' (Z + w + w + r) - i^. Substituting the values thence derived for S and S' in «a" — 8 — S', we get Imnrs - a (Zm+...+ ^5) -7 — a' (Z + m + n + r + «) + 3i In other words, the diminution from the number Imnrs pro- duced by curve and surface together is equal to the sum of their separate diminutions lessened by three times the number of their common points. 290. This result may be confirmed by supposing one of the quantics to be a complex one Z'Z'\ where Z' contains the common surface, and Z'' the common curve ; and the degrees of Z\ Z' are /, /'. Then the quantics Z7, F, W, F, Z\ by Alt. 288, have common points not on the common surface Z7nnr/-a{/(Z+w+&c.) + Z7w+ww+&c.}4-/8(/+Z+7n+&c.)— 7. But among these will be reckoned the aV points in whicli the common curve meets Z\ deducting however the t points common to the curve and surface. To find then the number pf points TJVWYZ' which lie on neither curve nor surface, we must deduct from the number last written aV — t. Consider now the intersections of Z7, F, IF, F, Z" ; these are a system of quantics having common two curves inter- secting in i points ; viz. the given curve a', and the curve of intersection of the common surface by Z'\ whose order will be flw'', and whose rank will be ow'' (X + ft + v + O* The number of points TJVWYZ^' which lie on neither curve nor surface will be . Imnrs''- (a'4- a«'0(^ + m + rt + r 4 0+ i8'+ a5"(X+/i* + v+O + 21* . QUANTICS WITH TWO SURFACES COMMON. 805 Adding, and writing s for / + /', we get Im^irs - a {Im +* * .+ ^«) + &c. ^ as in the last article. 291. We next suppose the quantics to have common two surfaces having i points of intersection. The method would be the same if there were several surfaces. Let the last quantic be a complex one, consisting of Z^ which passes through the first surface and Z^'^ which passes through the second. Then the system J7, F, Wy Y^ Z\ have the common surface \ilv and the curve XfiWs^ which have t points common, and the number of pomts of intersection, not lying on either surface, is thus . Imnra^ — X^v {(Z+ m + n + r) 5' + Zm + &c.} -f )8 (Z+ m + w + r + /) -7-X>V/(Z+wi + w + r + «') +>.VvV(\' + / + / + «') + 3*- In like manner for the system ?7, F, TF, F, Z^\ the number of points of intersection not lying on either surface, is Z7Wwr5"-X/Ay5"(7 + wi + «+r-f O + V^^^^C^ + A^ + ^-^O -X>V {[l + m -f w + r) /'+ ZW + &C.1 + ff (Z+ wH-w+r+/') + 3i. Adding these, we have for the whole number of points of intersection Z7wwr(«' + O-(V^ + ^V0{(^ + wi + w + r)(5' + «'') + Zm + &c.) + (/9 + i8'){^ + «» + w + r + / + 0-^7-'/ + 6«- In other words, the combined effect of the two surfaces is equal to the sum of the effects of the surfaces separately con^ Bidered, diminished by six times the number of their common points. When there are only four variables, two surfaces always must have common points of intersection. 292. Lastly, let the two surfaces have a common curve whose order and rank are a"j iS". Proceeding, as in the last article, we find that the system ZTFTFF^' have conlmon indeed the surface X/[av, and the curve Wfiy$. But since this curve is a complex one, consisting in part of the curve a", j8" which lies on X/ttv, we are only to take into account the (K)mplementary curve which, by Art« 382^ .has for its grder 306 ORDER OF RESTRICTED SYSTEMS OP EQUATIONS. XV V* — ^\ while its rank i8 i8" + (\>W - 2a") (V+ / + /-f 0; and the complementary curve intersects a'^^' in a^' (V + /*' + v' + /) - /3" points. The number of intersections is therefore Imnrfl — Xy^v {(Z + m + w + r) »' + Zw» 4- &c.} + )9(Z+?/i+n+r+«')-7 -(\>V/-a")(?+m + n + r + «') 4 /3" + (XVvV- 2a")(V4 /+ /+ /) + 3a'' (V+ /*'+ •+ d) - 3/3". Similarly the intersections for TJVWYZ^' are Imnrs"- X>V{(Z+7n+n+r)«"+ Zw + &c.} + yS' (Z+ w+ w +r+/0 - 7' - (X/iv/'- a") (? + wi + n + r + /') + /3"+(X/iv/-2a")(X + A* + v + + 3a''(X + ;A-fv+0-3/y\ Adding, we have Imnr (/ + «") - (X/iv + X'/iV) {(Z + w + n + r) («^ 4- «'0 4/^4 &c.} 4(i84/8'4 2a'')(Z4w4n4r4«'4 -7-7'4a"(X4M + v4X'4At'4/)-4^'. In other words, the diminution is obtained by regarding the two surfaces as making up a complex surface, whose order is the sum of their orders, whose rank is the sum of their ranks increased by twice the order of the common curve, and whose class is the sum of their classes increased by four times the rank of the common curve and diminished by xx''(X4/i4v4X'4/4vO- We must leave untouched some other cases which ought to be discussed in order to complete the subject; in particular the case where the surfaces touch in points or along a curve. 293. We come now to the problem of fipding the order of the system of corfditlons that three ternary quantics should have two common points. The method followed is the same as that given by Prof. Cayley for eliminating between three homogeneous equations in three variables, and which we have explained (Art. 94). Let the three equations be of the degrees Z, 7w, n. Multiply the first by all the terms a?*"""""", y»*^""*, &c RELATIONS AMONG THE EQUATIONS. 307 of an equation of the degree m + n — 3, the second in like manner by all the terms of an equation of the degree w + Z — 3, and the third by all the terms of an equation of the degree ?+ w — 3. We have thus in all ■J (wi+n-l) (rw+n-2) + i (w+Z-l) {n+l-2) + ^ {l+m-1) (Z+ m - 2) equations of the degree Z + w + w — 3, from which we are to eliminate the J (Z + 7w + w - 1) (Z + w + n - 2) terms oj^"'"^""*, &c. But, as it has been shewn in the place referred to, the equations we use are not independent, but are connected by i(Z-l)(Z^2)+i(w-l)(w-2) + i(«-l)(n-2). relations. Subtracting then the number of relations, the number of independent equations is found to be one less than the number of quantities to be eliminated ; and we have a matrix in which the number of columns is one more than the number of rows, the case considered in Art. 272. But, as was shewn, Art. 93, when we are given a number of equations connected by rela- tions, the determinants formed by taking a sufficient number of the equations, require ta be reduced by dividing out extraneous factors, these factors being determinants formed with the co-* efficients of the equations of relation. If then, in the present case, we took a sufficient number of the equations and deter- mined the order by the rule of Art. 272, our result would require to be reduced by a number which we proceed to determine. 294. Let us commence with the simplest case where we have k equations in k variables, the equations being connected by a single relation. To fix the Ideas we write down the system with three rows a, 5, c \ «', i', c' \' a", i", c" X" where we mean to imply that the quantities involved are con- nected by the relations Xa + XV + W = 0^ X6 4- X' J' + X"6" = 0, Xc + XV + X'V = 0. We also suppose that Xa, XV, X'V are of the same order, so that the orders X + a = X'-f a' = X''+ a^\ Now let us, In the first SOS OSPEB OF R£8TRICT£P SYSTEMS OF EQUATIONS* place, suppose that the two first, equations are in the simplest form, and that X^^ s — 1. The true order then is that determined from the first two equations ; that is to say, if we indicate the orders, as in Art. 272, (7,+0, (a4-i8)H-a*+i8*+a^, Now suppose that we had omitted the first row, the order deduced from the second and third would be C, +(7, (iS + 7) + )8* + 7* + ^87 ; which we see, in order to give the true order, requires to be reduced by (7 — a) ((7, + a + i8 + 7) ; in other words, by the order of \ multiplied by the order of the determinant obtained from the three equations. And the general rule to which we are thus led is : Leave out one of the rows and determine the order of the remaining system by the rule of Art. 272 ; from the number so found, subtract the order of the determinant formed from all the equations, multiplied by the order of the term in the relation column belonging to the omitted row. It is easy to verify, that we are thus led to the same result whatever be the omitted row. Thus C; + (7,(a + fl) + a' + /3' + a/8-V'((7, + a + )8 + 7) since the orders \ — X" = 7 — a. And our result may be written in a symmetrical form if we write A for the common value of X + a, X' + ^8, X'' + 7, when it becomes C;+(7,(a + )8 + 7) + a'+i8'+7'+i87+7a+ai9--4((7,+a+/3+7), pr C,+ 0,5^ + 5,-^ (0, + S;), 295. And, generally, if there be any number of relation columns, I have been led by a similar process to the followbg result : Let the terms in the relation columns be A, X', X'', &c^ /A, f/j yi!\ &c., V, v', /', &c. ; then we must have X + a = X' + /3', &c., /iA + a = /i*' + )8', &c. F + a = i/ + i9', &c. Let A^ Bj C denote the common values of these sums, and let ^/, J7/ denote the sum and sum of products as in Art. 272 of the quantities A^Bj C] then the order of the system is This result may be stated as follows, in a way which leads us at once to' foresee the answer to some other questions that TWO POIKTg COMMOK TO THR££ TERNABY • QU ANTICS. 809 may' be proposed as to the order of systems of these equations.' in the case we are considering, the entire number of columns/ counting the relation columns, is one more' than the number of rows ; and the order of the system is that given by the rule of Art. 272, if we give a negative sign to the orders in the re- lation columns. In like manner, when the number of columns, coimting the i'elation columns, is equal to the number of rows,' the system, by Prof. Cayley's theorem, represents a determinant whose order is that which we should obtain by calculating the order of the entire system considered as a determinant, the orders in the relation columns being taken negatively. And so no doubt if the entire number of columns exceeded the number of rows by two, the order of the system would be found by the same modification from the rule of Art. 285. 296. Let us now apply the rule just arrived at to the problem proposed in Art. 293. We consider the three ternary, quantics of the order Z, m^ n respectively ; and we regard these as depending upon two arbitrary parameters, the orders in these parameters being as follows; the coefficients of sf^s^^ a;% the highest powers of a;, are of the orders \, /*,,f; those of ^'^Hy ^'^^ aro of the orders \ + a, \+a', and so on, the orders of the coefficients increasing by a for every power of y, and by a' for every power of z. Then the terms in the first column consist first of ^(m + n — 1) (wi + n — 2) terms whose orders are X; \-a, X — a'; X — 2a, X — a — ot', X — 2flt', &c.; secondly, of i (w + Z— 1) (n + Z— 2) terms whose orders are fi] /Lt — a, /i* — a' ; &C.J and thirdly of ^(Z + wi — l)(Z + m — 2) similar terms in v. These may be taken for the numbers a, iS, 7, &c. of Art. 272. The numbers a, i, c, &c. of that article are 0, a, a' : 2a, a + a^, 2a', &c., there being in all ^ (Z + w + n — 1) (Z + »i + n - 2) suet terms. Lastly, the numbers A^ j5, (7, &c. of the last article are found to consist of J{Z— 1) (Z— 2) terms, fi+Vj fA-\-v—a, /Lt-j-v-ot'; together with J (w — 1) (m — 2) and ^ (n — 1) (n — 2) correspond- ing terms in f+X and X + /a« In calculating I have found it convenient to throw the formula of the last article into the shape where s^ denotes the sum of the squares of the terms a, b^ CySui. 310 ORDER OF RESTRICTED SYSTEMS OF EQUATIONS. Also if 4>{l) = Al'-\'BP+CP'^Dl + E^ it is convenient to take notice that ^(Z + m + n) + 0(Z)-f^(wi)+0(n)-0(Z+»i)-0(wi+n)-^(n+Q = l2Almn (Z + m + n) + OBlmn + E. I have thus arrived at the result, that the order of the system or number of the sets of values of the parameters is imn [mn - 1) X'+^l {nl - 1)/*" + ^Im {Im - 1) i/" + {(fiZ-l)(Zm-l)-i(Z-l)(Z-2)}/iy + {(Im • 1) (f/in - 1) - iCw - 1) (w - 2)} v\ + {(wZ-l)(fifn-l)-i(n-l)(n-2)}X/t4 + fnn\{lmn - Z+ 1 - ^ (m+n)} (a + a') + filfA {Imn - m + 1 - i (n + Z)} (o + a') + Imv [Imn - n + 1 -^ (Z + m)} (a + a") + ^Imn {Imn- Z— m— n+2) (a*+ a'*)+^lmn{2lmn — Z— «— n + 1) aa\ If the order of all the terms in the first equation be \, in the second /a, in the third v, we have only to make a and a^ = in the preceding formula. In this case, supposing X = /x = v = 1, the order becomes i (mn + wZ+ Im) [mn + nl+ Im — 5) -^(l-l)(l.2)-^{m-l){m-i)-i{n-l){n-2), aiid in particular if Z= m = n, the order is fn(w-l)(w' + n-l). This last result shows that if U, U\ W\ F, F', V'\ TF, W\ Tf " be given homogeneous functions of («, y, z) each of the order «, then the number of curve-triplets U-i-eU'+t^W'^O, F+eF^+0F'' = O, TF+eTF'+0TF"=O, having 2 common points, is = in(n-l)(n'' + w-l). 297. Mr. S. Roberts applies to the problems of this Lesson a method directly applicable to binary quantics, since they can always be resolved into factors, and which extends to the case of ternary and higher quantics, for the question whether or not they can be so resolved does not affect the METHOD OF MR. 8. BOBEBTS. 311 problems here discussed, and the orders determined in the case of quantics which ' are the products of factors must be generally true. Thus, to determine the order of the resultant of two binary quantics of the degrees m, n; if the order of the terms in the first be X, \ + a, X + 2a, &c., it may be resolved into the product of m factors ox + 5^, the orders of a and b being — , — Ha respectively ; similarly, for the second quantic; and the resultant is the product of mn factors, the \ X' ' order of each being — I ha: and, therefore, mn times m ft this number will be the order of the resultant. Now Mr. Roberts argues that we may deal in the same manner with the problem in Art. 277; that knowing, by Art. 272, the order of the matrix a, h to be a, h «', b' «", b" a* + (X + /Lfc + v) a + X/A + AAV + vX, the orders of the rows being supposed to be X, X + a ; /it, /A-f a ; v, V -I- a ; then we may conclude that the order of the system of conditions for the simultaneous existence of three equations of orders Z, »n, n is lmn\a*-ha(Y-h- + -) + 7^ + — + ^| . ( \l m nj Im mn nl) And in like manner, that the order of conditions for the co- existence of a system of & + 1 binary equations is the product of their degrees multiplied by where JP,, P,, &c. are the sum, sum of products in pairs, &c. of the numbers 79-^9 &c. And so more generally, the order of the conditions for the co-existence of any number of equations in any number of variables is derived from the order deter- mined by Art. 272 for the co-existence of a system of linear equations. It is thus found that the order of conditions for the co-existence o{ k-\-s— 1 homogeneous equations in s variables, in which the order of the coefficients of a;', x^y^ «:''*«, &c. is 312 ORDEB OF HEfiTRlCTED StSTEMS OF EQUATIONS. X) X + o, X + iS, &c. is the prodact of their degrees multiplied bj where S^ has the same meaniDg as in Art 272, and P, Qj &c. are the sum, sum of products in pairs, &c. of the numbers •J J —J &C. Thus, for instance, this formula applied to the case of ternary quantics gives the order of the conditions that a curve should have a cusp« We determine by the formula the order for the co*existence of Z7j, C^, t^, U^^ U^ - D^^*, which system belongs either to cusps or double points on the line z^ and we subtract the order for the coexistence of Z7,, C^, U^j z^ which belongs to the latter. The result is 12(n-l)(n-2)X*+8n{n-l)(n-2)(a + i8)X + 2«(«-l)(n-2)(n+l)oi8 + 2n(«-l)'(n-2)(a* + i8'). The problem of finding the order of conditions that two binary equations should have two common roots is discussed as follows : Consider first the simpler system, formed by taking two factors from each equation, {ax + hy) [a'x + Vy) {cl'x + V'y) [a'^'x -f y'», and we have the pair of conditions (aJ'O [a'V) = 0, [aV') (aT'). = 0, whose order combined is 4 (X + /* + a)* ; but from this we must subtract the irrelevant systems {aV) [aV'X («'0 («'i"0> which reduces the order to 2 (X + /AH-a)'. But if we take two factors from the first equation and one from the second, the system (ai'O = 0, (a'O = is satisfied by a'' = 0, V = 0, whose order is /L6 (/L& + a). Now since the number of ways in which two factors of the first equation may be combined with two of the second Is ^Z(Z— 1) x i7/i(w — 1), and the number of ways in which one of the second may be combined with two of the first is ^Z (Z— 1) 771 ; the resulting order in general is as in Art. 275. METHOD OF MB. S. ROBERTS. 313 ' . By tbe same process of reasoning Mr. Koberts arrives at the order of the conditions (Art. 296) that three ternary quantics should have two points common, in the form {ilran{l'l) (m-l)(«-l)+SiZ7wn(m-l)(w-l)} j^ -f i!? + - +a+a'| + sji„. (r- 1) {^* + £4 + ^ + (. + ^)(^ + e) w} . In this way the order of conditions that a curve should have two double points is found to be i(n-l)(«-2)«(n+l){3X + n(a + a')}"-i(n-l)(n-2) . X {15\" -h lOn (a + a') X + n (w + 6) aa' + 2n (2n - 3) (a* + a'*)}. Mr. Boberts investigates other problems by the same method ; as, for instance, the order of conditions that four curves may have two points common, or that a surface may have a biplanar double point. For these I must refer to his paper.* I only give the following result: The order of conditions that three binary quantics should have two roots common is +^i'"»c»-')(»-i){r(7 +<■)(£ + ;+«)} * Prof. Cayley, in the Cambridge and Dublin Mathematical Journal, vol. iv., p. 134, determined the order of a matrix with k rows and ^ + 1 columns, in the particular case where each constituent is of the first degree. My own investigations were pub- lished, Quarterly Journal, vol. I., p. 246, and in the Appendix to my Geometry of Three Dimensions, second edition. After this Lesson w^ printed in the second edition Mr. Samuel Roberts communicated to me some extensions of the theory there developed, and his results have since been published, Proceedings of the London Mathematical Society, March, 1875. SS ( 314 ) LESSON XX. APPLICATIONS OF SYMBOLICAL METHODS. 298. In this Lesson, which is supplementary to Lesson XIV., we wish to show how the symbolical notation there explained affords a calculus by means of which invariants and covariants can be transformed, and the identity of different expressions ascertained. In order to facilitate the reader's study of recent memoirs, we employ the notation explained, Art. 162, which is now almost exclusively used; to save the necessity of reference, we repeat what has been already said, and, in order to fix the ideas, we suppose the variables to be three, though the method is perfectly general. The variables then are a?„ iCj, a;,; if there are different sets of cogredient variables, such as the coordinates of different points, they are written Vi) Vit l/ij ^vt ^2? ^8J ^^' ^^ ^^® ^^^ abbreviation a^ for a^x^ + a^x^ + a^x^, a^ for a^y,^ a^y^-V a^y^] if we are only dealing with one set of variables so that no confusion is likely to arise, we sometimes suppress the suflSx, and write a instead of a^. The quantic of the n^^ degree is symbolically written ^x'i ^^ (<^i^i + ^2^1 + ^z^S ; ^^^^ ^^ *^ say, ttj, a^, a^ are umbral symbols not regarded as having any meaning separately ;* but a^ denotes the coefficient of ar," in the quantic, a^'^a^ that of nx^'^x^^ and so on. And so generally any homogeneous function of the rfi^ degree in the letters a^, a„ a^ may be replaced by a multiple sum of the coeflScients of the quantic; any other function of these letters is not regarded as having a separate meaning. Other quantics may be denoted by 5^, c', &c., the symbols J^, J^, b , &c. being used in the same way. In the cases with which we principally deal the quantics are supposed to * It has, however, been stated. Art. 163, that we can at any moment interpret a formula by substituting for Oj, Og, a^ differential symbols with regs^rd to ar„ a:^, or,. SYMBOLS FOR CONCOMITANTS. 315 be identical ; and a", i", c", &c. are onlj diflferent expressions for the same quantlc. We use {abc)j {abd)j &c., to denote determinants formed with the constituents a^^ a„ a^] i„ &„ i,, &c. In order to express invariants or covariants of the quantic we take any number of such determinants and multiply them together 5 then evidently the product can be translated as a function of the coefficients of the given quantic, provided that the a symbols, b symbols, &c. respectively each occur n times. If not, we join in the product such powers of a^, J^, &c, — that is to say, of {ctiX^ + a^x^-^-a^x^j &c. — as will make up the total number of a's, 6's, &c. to w. We are then able to replace the symbolical letters by coefficients of the quantic, and the resulting product is a function of the coefficients and the variables, the latter entering in a degree equal to the sum of the orders of a^, 6^, &c. in the symbolical product. It is easy to show that we obtain an invariant in the one case and a covariant in the other j and we refer to Clebsch'sl Theorie der binaren algebratscken Formen for a formal proof that all invariants and covariants can be so expressed.* All this has been stated already (Art. 162). When a covariant is expressed in the manner explained, it is evident that its order in the coefficients is equal to the number of symbols a,, i„ &c., which enter into the determinant factors, and that its order in the variables is equal to the number of non-determinant factors a^, J^, &c. Since the diflFerential coefficients of w = «/ ^^^ respectively na^'^a^^ na^'^a^^ na^'^a^^ the equation of the polar, which is K^' £ "^ ^' £ + ^» £' ) = ^' ^'^'^^'' "'"""' ^ ^- S''"^^*^'^' X ^ 3 the second polar is cL^'^a^^ and so on. ------ ■ * The principle of the proof is briefly this : we have seen, Art. 21'8, that from any d invariant or covariant P of a single quantic, we can, by the operation a' -j- f &c., da obtain a corresponding form 11 for a system of two quantics, and that we can fall back from n on P by making a' = a, &c. By repeated application of this principle^ if the form P be of the r** order in the coefficients, it may be considered as derived from a form II belonging to r different functions, each of which may be symbolically written (fljo;, + flyCa)", (^i^Bi + ^2^)", Ac. Every form therefore of the r** order for the single quantic has a corresponding form for r linear factors, and it is proved without dif&culty that for the latter case the only invariant or covariant forms 91& those expressed as in the text. dl6 APPLICATIONS OF STMBOUCAL METHODS. 299. ConfiDing ourselves now to the case of two variables, a^ or a here stands for OjX^ + a,a?„ {ah) stands for afi^ — ajk^ 5 and any covariant is expressed symbolically by a product {ahy (ae)fi {bdy &c multiplied by o'ftV &c., the number of a's h\ &Cj in the entire product being each n or a multiple of n. If /?, q^ r, &c all vanish, the symbol denotes an invariant. Any symbol which simply changes sign by an interchange of a and b (as, for example, [abyci^'V^^ where a is odd) denotes aii expression which vanishes identically (see Artr 153). If we eliminate a?^, x^ from the equations we have {A) a(Jc) + 6(ca)+c(ai) = 0, an identity of the greatest use in transforming these expressions. Thus, for example, transposing a{]bc) to the other side and squaring, we have (B) 2Jc (oi) (oc) = V (ac)« + c* [aVf - a' [bcf. To illustrate the use of this, multiply by a'^b'^c''^^ in order that each term may denote a covariant of an n-ic, and we have 2a" '^b'^c^ [ab) [ac) = &"a"-*c"-' {acy+ c'^a'^'b'^' [aby - oTb'^c'"' (bey. Now, since a", &", c* all equally denote the quantic, the three terms on the right-band side of the equation are only different expressions for the same covariant; and we learn that the covariant cT^b'^'^c^^ [ab) [clc) is half the product of a* (which Is the quantic itself) by b*^c^^ (Jc)*, which denotes the Hessian. We can always (as has been stated Art. 163) interpret these symbolical expressions by supposing a,, a„ &c., to denote T— , -f-j &c., by supposing that we operate on the product of several distinct quantics uvtOy and by making the variables identical after dIflFerentiatlon. In this way a or x^ -= — \- x^-j-j applied to any homogeneous function, only affects it with a numerical factor. If we thus interpret equation (£), {ah) {a^ FORMUKfi OP TRANSFORMATION. .i317 i» Q^ where Q is dx^ \dx^) dx^dx^ \dxj \dxj dx* \dxj ' and (abV is 2H, where H is -^—5 --7—= — ( -^ — ^— \ . ^ ' dx^ dx^ \dx^dxj On the right-hand side of equation B^ a', J', <? respectively operate on functions which have not been before differentiated, and therefore affect them with the numerical factor w (w — 1)'; on the left-hand side &, operate on a function which has been once differentiated, and each affects it with the numerical factor (w — !)• Equation B therefore gives us (n — 1) Q = w UH, 300. From equation B other useful formulae may be derived. Squaring it, we have {G) a*(Jc)* + J*(ca)* + c*(aJ)* = 2 {6V [ahy [acf + cV {he)' {laf + a^V {caf \cbY]. If this be applied to a quartic form, the three terms on the left- hand side are all different expressions for the same thing. So are also the three terms on the right; and we learn that the covariant iV [ah)' {cicf differs only by a numerical factor from the product of the quantic itself by the invariant (ic)*. Considering the four expressions we at once verify the identity (D) [ad) {he) + {hd) {ca) + {cd) {ah) = 0, which is also of very great use in the theory. We deduce from it {E) 2 {hd) {cd) [ah) {ac) = {hd)' {ac)' + {cd)' {ah)' - {ad)' {he)'. {F) {ad Y {he)' + {hdy {ca)' + {cd)' {ah)' ^2{{hd)\cd)'{ah)'{acY^cd)'{ad)'{bc)\ha)'-^{adY{bd)'{ca)'{cb)']. To these may be added an identity which is really a different form of {B) : this is iO) %K^hji^={ah){xy), where {xy) denotes {x^y^ - x^y^ ; and from it we deduce 318 APPUCATIONS OF SYMBOLICAL METHODS. 301. A symbolical expression may be always so transformed that the highest power of any factor {ah) shall be even. For the signification of the symbol is not altered if we interchange the letters a and b ; therefore , and by the help of equation A^ ^, — <^, can always be so trans* formed as to be divisible by (ab). Thus ic*""' (aJ)*"*^ {ac) is at once reduced to c*" (ai)*". For Jc*-* {ab)"^' [ac) = ic*-* {abr-' {b [ac) -a{bc)}=^ ic«"' (a&)«". 302. If we arrange symbolical products according to the number of determinant factors which they contain, we can, by these formulas of reduction, reduce them to certain standard forms. If there is but a single factor (ai), the covariant vanishes identically (Art 299), since it changes sign by an interchange of a and b. The possible forms with two factors are (a6)*, (a6)(ac), (a6)(cJ), of which the last vanishes identi- cally, and, in Art. 299, we have expressed [ah) [ac) in terms of [ab)*. There is therefore only a single distinct covariant symbolically expressed with two determinant factors, viz. the Hessian fi^= a""*6""* (aij*. Any such form must denote either the Hessian or the product of the Hessian by a power of the quantic. So again, for three factors, the possible forms are [ab)\ [ah)''[ac), [ab) [ac)[ad), [ab) [ac) [be), [ab) [ac) [bd) ; of these the first, fourth, and fifth vanish identically, and the second and third are found to be related as follows: Multiply equation B by a~^b'''^c''"^d''~^[ad) ; two of the terms on the right- hand side become identical, the third vanishes, and we have 2a""'i""Vd"-'(aJ) [ac) [ad)=2a'^%'"^-c''d''-\abY [ad), showing that the covariant expressed by the left-hand side is the product of the quantic itself by the covariant whose value is given (Art, 156). Generally every symbol having a pair of factors with a common letter [ab) [ac) may be reduced to a more compact form by substituting for this pair their value from equation B, and so expressing the symbol by others in which this pair of factors is replaced by a single square factor. Symbols with four factors can be reduced to either of the forms REDUCTION TO STANDARD FORMS. 319 {aby or (a&)*(crf)', which is R*. For five factors the fundamental forms are (aJ)*(ac) and {abY{ac){deY, the latter being the pro- duct of two distinct covariants. For six factors the forms are (aby (Art, 153), {abY{bcY{cay (Art. 155), and [abf [cdf {e/)\ the last being the cube of the Hessian. We give a few examples of the reduction of these forms. Ex. 1. To reduce a»-**"->c»-»«^»-»6«-i {ab) {ac) (ad) (ae). Multiply together 2bc {ab) (ae) = b^ {acf + C^ {bhf - a* (ic)*, Ide {ad) {ae) = d^ (ac)* + «« {ad)^ - o« {de)^ ; multiply the product by a*-*b*-^c*-^d**~^e*-^, and assemble the identical terms, then we have 4aii-4^-i^-i^n-ign-i (^) ^ac) {ad) {ae) = 4<?»e"a"-<*»-»(i"-2 (a*)2 {ad)^ - 3a«ft»-2c"-2<f "-««"-« (ic)« (de)« The last term is — QUH^. The other term on the right-hand side is reduced by equation C, Multiply by o«-*d»'<e?«-* the equation a* {bd)* + b* {ad)* + d* {ab)* = 2 {b^-d^ {ab)^ {adf + d^a^ {bd)^ {bay + am {adf {bd^}, ' then we have 2a«-**'»-2(?«-2(ai)2(a(i)2 = rf»»a"-*A»-*(aft)*. The right-hand side there- fore of the preceding equation reduces to 18U^ -^ ^UH\ where 8 is the covariant a*-*b^-*{dby, Ex. 2. To reduce {ab)*{acY. Multiply equation C by a^'i-'^c" *(a6)2, then we have, assembling identical terms, c"o»-«*"-« {obY = 2a"-«6«-*c»-2 (aft)* (ac)* + 2a"-«6'»-«c"-'* (oJ)* (6c)* (ca)*. Thus, if we call the standard forms a»-«A"-« (aft)« and a"-**'*-*^"-'* (aft)* (ftc)* (ac)*, Jf and r, we have 2a»-«6»-«c'»-* {ab)* {ae)^ = C^J/- 2r. Ex. 3. To reduce (aft)* («c)* (a<f )*. Multiply together the three equations 2ftc (aft) (ac) = c* (aft)* + ft* {ac)^ - a* (ftc)*, 2crf(ac) {ad) = e?* (ac)* + c* (aef )*- a* (c<f )«, 2ftrf (aft) (a<?) = ft* (arf)* + c?* (aft)* - o* (ftc?)*, and multiply by a""*ft'»~*c'*"*rf*»~*, then we have 6a«-«ft'»-*c«-*<?"-* (oft)* {acf {adf = - 4(i'»a»-*ft'»-«c"-'*(aft)*(ftc)*(ca)*+6o'»-«ft«-'*c"-*c?"(aft)«(ac)2-3a»-*ft'»-«c"-2(i«-2(aft)* (crf)« = ~ 4UT+BU{UM- 2T) - 35i7 = 3£72Jf - lOUT-dSH. Ex. 4. To reduce (oft)* (ore)* {cd)^. Multiply equation {C) by (erf)*, and by un-4^n-4^-6(^n-2^ then WO have 4<2«-4jn-2c»-4e;«-2 (aft)*(oc)* (erf)* - 2a«ft'»-«c»-6rf»-* (ftc)« (crf)*+ a»-4jn-4^2ef »»-« {ab)* {cdf - 2a"-*ft'»-*c'»-«rf»-* {acY{bcY{dcY Multiply by three, and observe that in Ex. 3 we find that 6a«ft»-*c»-«rf»-* {be)* {cd)^ - 6a»-*ft»-*c'»-«rf'»-* {acf {bcf {dcf = ^UT + MH, «nd we have 6a»- 4jn+2cn-4^n-2 (^ft)* {acf {cdf = 2 CTT + 355: Ex. 5. To reduce (aft)* (erf)* {ac) {bd). Multiply the equations 2arf (erf) {ca) = a* (erf)* + rf* (ac)* - c* (arf)*, 2ftc (erf) (ftrf) = ft* (erf)* + c* (ftrf)« - rf* (fte)*, 820 APPUCATIOMS OF SYMBOLICAL METHODS. «Dd multiply also by a*-«d*-V*~^<{*-^ (o^)', and we hare - 4a!»-»6^V^»rf«-» (a6)« (erf )« (ck?) (M) - 2a^6»-*tf»-*rf» (a*)« (*c)« {caf zzSH- 2TU+ 2«»-*ft«-«<f^d«-« (a6)« (crf)« («?)« ; therefore, by Ex. 4, - 12a»-»*^<j^d»-« (a*)» (crf)« (atf) (W) = Gflffl^ - ^TU. We stated (Art. 156) that the expression for the discriminant of a cnbic was {ab)* {cd)* {ac) {bd). The present example shows how the corresponding oovariant of a qnartic can be expressed in terms of the fondamental forms 8^ T^ U, H, 303. If ^, ^ be covariants of the orders p, q in the vari- ables, we may write these symbolically ^ = [4^\^i + 4>%^^i •^ = (V^,a?i + ^2^2)*^ *°^ ^® ^*^ obtain from them the series of covariants <f>J'^'^^{4>'^)^j where k has any value from 1 up to the least of the two numbers p and q. This operation, in German called Ueberschiebung^ we shall call trausvection, and the covariants generated we shall call transvectants of the two given covariants. In the notation used Lesson XIY, 12^^^ denotes a covariant diflfering only by a numerical factor from the transvectant; that is to say, if we denote differentiation with regard to a?^, x^ by subindices, the series is Thus the first in the series is the Jacoblan, and if n be the lower of the orders of if> and yfr^ the last in the series 12*^^, is the result obtained by introducing differential symbols into the one quantic and operating on the other (Art. 139). We obtain transvectants of a single quantic by supposing <f> and ^ to denote the same quantic. The transvectants then of odd order vanish, and those of even order form the series of co- variants considered (Art. 141). The method of formation of covariants explained in this article has a prominent place in the proof given by Gordan and Clebsch, that the number of covariants of any form is always finite. Thus the first step in the proof is to show, as we shall presently do, that any covariant symbol formed with k letters a, J, c, &c. may be reduced to a transvectant of the original form combined with a covariant whose symbol contains only A: — 1 letters. TRANSVECtlON. '321 304. There ib no diflSculty in forming the transvectant, or any other derivative of a form <f> symbolically expressed. Thie transvectants of ^, ^, we have seen, are given by the formula {^^)* = {^^Vt "" f t^i)*» where f j, f , denote differentiation of 0, and ^p ty, differentiation of '^. Now let the symbolic expression of be = lfa'*i', &c., where M denotes the aggregate of the determinant factors. Then^ since x^^ x^ only enter in a, &, &c., we have Similarly if ^ = NifcC &c , we have If we put these values into (^^17, — f ,'7i)* we see that the symbolic expression for the transvectant consists of a group of term^, each of which contains all the determinant factors Jf, N of the two given forms, together with a function of the k^ order in the determinants (ac), (arf), (Jc), (Jc?), &c. In particular the transvectants of <f> combined with the original form u (or 9**, where ^ is a symbol not occurring in 0}, are found by operating on * with ^{ag) ^ + [hg] ^j + (<^) ^ + &c j . Ex. To form the symbol for the Hessian of the Hessian. Here we haye ^ = o»-26»-2(a6)«, ^ = d*-'^d^\cd)^, and we are to operate on <^^ with (fjij^ - ^^»yj« j or with * or, collecting terms, with It would be therefore 4 (n - 2)« (n - 3)« (a*)« (cd)* (ac)« a"-4jn-2c«-4^n-2 + 8 (« - 2)» (« - 3) {aby («?^2 (am?) (arf) a«-«i»-2c« »(?•-» + 4 (» - 2)* {abf {cdf {ac) {bd) a»-»i»-3c"-3ei"-». It was shown, Art. 302, Ex. 4 and Ex. 5, that the first and third of these terms were expressible in the form aSH + ^TU, and the same thing is easily seen to be true of the second, if we substitute for ^cd {ac) {ad) its value c^ {ad)'^ + d^ {acf — a^ {cd)^ from equation B. Thus we prove that the Hessian of the Hessian is expressible in the form aSS + fiTUj as was otherwise proT€d (Art, 211)* TT 322 APPLICATIONS OP SYMBOLICAL METHODS. 305. It is convenient to transform the formula given in the last article for (u^), the transvectant of any form (p combined with the original. By equation A where it will be observed that the operator a-y + J -^ + &c. ' da do leaves the subject unchanged except by a numerical factor. Thus then if ^ = Md^lfd^ &c., where 2> + j + ^ + &c. = I {u<f>) = lM[ag) a^'h'c\.,g''-' ^g^'MoT' \{ab) ^ -4- &c. J V &c. Thus we have the expression divided into two parts, one in which we have the old determinant factors M together with the new factor (ag) ; the other in which the letter g does not enter into the determinant factors. Conversely, if we have a symbol in A; + 1 letters, one of which g only occurs once, this is immediately resolved into the transvectant of 0, u^ together with a form whose symbol only contains Jc letters. So, in like manner, the second transvectant with any form (f> is {u4>y = |(a^) ^ + {Ig) ^ + &c.|' <^, and therefore consists of a group of terms, each having all the determinant factors of ^, together with two new factors, each containing g. But, as before, this might be written and therefore if ^ = Ma'^b^ &c., we have {u<l>y = Z (Z - 1) M[agyar'V...g''^ + ^, where y^ consists of a group of terms into the determinant factors of which g enters only once ; and therefore by the former part of this article, '^ can be reduced to a function whose X REDUCTION TO TRANSVECTANTS. 323 syrabol does not contain g^ together with the transvectant of such a function. Thus then any symbol of the form <f) (agy can be reduced to a function whose symbol does not contain g^ together with the first or second transvectants of such functions. The same thing would be true of a function of the form 4> iP'O) (p9)% as appears by writing the second transvectant in the form And so we see generally, that any symbol <^ («.^)' C^J)^ where g occurs in all k times, can be reduced to the A'** transvectant of 0, together with terms in which g occurs only k — 1 times, which again may be reduced in like manner. If, then, we arrange forms according to their order in the coefficients, it has been proved that the forms of any order consist either of forms of lower order multiplied by u or by powers of w, or else of transvectants obtained by combining u with forms of lower order. 306. We have just shewed that taking any one letter g in the symbol for a form, that form may be regarded as a transvectant of u combined with other covariants, and that the symbol for each of these other covariants will contain all the determinant factors of the original form, striking out those which contain g. So again taking any letter f in the symbol for any of these other covariants, we express that covariant as a transvectant of u combined with covariants whose symbol contains all the determinant factors of the original, striking out those which include the letters f and g. Proceeding thus we see that taking any of the letters, say, a, 6, c, and considering the factors of the given form which contain these letters only, say [ahy [bcY (ca)"", then the given form may be obtained by transvection from a form in these three letters having as a factor {ahy {bey {ca)\ All that has been said in the last article applies equally if the original form, instead of being a covariant of a single 324 APPLICATIONS OF SYMBOLICAL METHODS. . function, were a simultaneous covariant of several; that is to say, if instead of a", &*, c* all representing the same function U^ they represent different functions 27, F, Wj &c. It remains true that if V be the function to which g refers, the form represents a transvectant of V and of other functions, whose symbols could be separated in like manner. It may be seen thus that if we had all possible co variants of two forms i[7, F separately considered, including in the series the powers and products of the simple covariants, the system of the simultaneous covariants o( Uj V is obtained by adding to these all possible transvectants of a form of one set with a form of the other. If the forms so obtained be combined by transvection with the covariants of a third fundamental form W^ we obtain all possible forms of the system Z7, F, TF, and so on. We refer to Clebsob, p. 186, or to Gordan, Prcjjrawim, p. 18, for a proof that if the system of covariants of the separate forms U^ F, W be finite, the number of distinct forms obtained by transvection in the manner described, and therefore the number of covariants of the system, is also finite. 307. We proceed now to give an outline of the method by which Gordan has shewed that the number of distinct forms for a quantic of the n^^ degree is finite, and we shall shew that if this be true for a quantic of the degree n— 1, it will be as true for one of the degree n. ' The symbol for a form belonging to a quantic of the {n- 1)^^ degree being MaFb^c^ &c., where the a symbols, b symbols, &c. each occur n — 1 times, it follows that if we multiply the symbol by abc &c. we shall have a form for a quantic of the n^ degree, since the a symbols, &c. will then each occur n times. We shall speak then of forms belonging to quantics of different degrees, as being the same, when the determinant factors in their symbols are the same, and when these symbols only differ by a power of abc &c. We propose to establish the following theorem, viz. that the forms for a quantic of the n^^ degree consist either of the forms which had occurred already for a quantic of the degree n- 1, or else of the mutual transvectants between such fornas and the series of two-lettered forms (ai)*, (a6/, &c. And in order NUMBER OF DISTINCT FORMS PROVED FINITE. 325 -to establish this, we shall shew in the first place that eveiy form for a quantic of the rfi^ degree either has in its symbol a factor {alf where p is not less than ^n^ or else is a form belonging also to a quantic of the (n — 1)^^ degree ; and with regard to the latter, we add the further restriction,* that if its symbol be Md^¥(^ &c., p the greatest of these indices shall not be less than \n. We shall prove this latter theorem by shewing that if it is true for forms of the ni^ order in the coefficients, it is true for forms of the order m + 1 ; and it evidently is true for forms of the first order, that is the quantic itself. Now it was shewn (Art. 305) that all forms of the order w -f I can be obtained from forms of the order m by transvection of such forms with u. Since this transvection only adds new determinant . factors to the symbol without removing any of the old, it follows that every form of the order m having {ab'f in its symbol, will give rise by transvection to forms of the order wi + 1 having the same property. We need only consider therefore forms Jfa^JV &c. Now we saw (Art. 304) that the 1^ transvectant of such a form is got by operating Tc times with that is to say, it will be of the form M[agfdr^Vc^..,g'^^ together with transvectants of the order A — 1. The term which we have written will, in case k is as large as ^n, contain the factor {ag)^] and if k be less than \n^ then since by hypothesis p is at least ^w, p- k the index of a, will be positive, and therefore the term will still be divisible by ahc &c., and will therefore denote a form belonging to a quantic of lower order, while n — k the index of g^ will exceed \n. We see then, that if forms of the order m in the coefficients consist only of the two classes we have named, the k^^ transvectant of such form will consist only of the same two classes, provided this be true for the (i - 1)^*^ transvectant. * This limitation is only necessary for the proof of the theorem, and is not used in )miy of its subsequent applications. 326 APPLICATIONS OP SYMBOLICAL METHODS. And SO down step by step till we come to the transvectant ; that is to say, the product of the given form by a, for which it is obviously true. The theorem we have enunciated is therefore proved. It follows from what has been just proved, that every invariant symbol must contain as a factor [abf^ where p is at least half n ; for the other class of symbols, viz. those occurring in quantics of lower order, of necessity represent covariants, since they have the factors a, 5, &c., each of which contains the variables. Forms having [aby for a factor are (Art. 306) transvectants of forms having this as a factor, and it can be seen without difficulty that, except when n = 4, such forms will when p exceeds \n be of lower order in the variables than the quantic itself. With this exception, then, it appears that forms for the fi*^ degree are simultaneous covariants of certain forms all of lower order than «; and therefore if the number be proved finite for numbers less than n, it is also finite for n (Art. 306). The case n = 4 requires a little speciality of treatment, for which we refer to Clebsch, p. 267. 308. It would evidently be convenient if a general symbolical expression could be given for the result of elimination betweea two equations. When one equation is simple it is easily seen that the eliminant between it and an equation of the n^^ degree is (aa) (a^) (ac), &c., where the symbol a relates to the equation of the n^^ degree, and the remaining symbols to the simple equation. I gave in 1853 a general formula for the resultant of a quadratic and an equation of the n^^ degree {Cambridge and Dublin Math, Jour.* IX., 32). The theorem was re- discovered by Clebsch in 1860, and extended by him to the * In the same paper I investigated a formula for the discriminant of a binary quantic, and in this way obtained that for a quartic. Clebsch subsequently gave in hia paper {Crelicy Lix.), "Ueber symbolische Darstellung algebraischer Formen," a rule for obtaining a general symbolic formula for the resultant of two binary quantics or for the discriminant of a binary quantic. The method of proceeding is to apply Cayley's form of Bezout's method of elimination, Art. 87, to two quantics written Bymbolically, but the resulting rule is, as might be expected, very complicated. ELIMINANT OF QUADRIC AND ANY QU ANTIC. 327 case of a system of any number of equations, one of the second , one of the n^^ and the rest of the first degree ( Crelle^ vol. LVlii). We give here Clebsch's investigation of the general theorem. To fix the ideas, I write only a system of four homogeneous equations in four variables, but it will be understood that the method is equally applicable to any number of variables. Let the equations then be a = a^x^ + a,ar, + a^x^ + a^x^ = 0, /3 = ^^x^ + ^,ar, + P^x^ + ^^x^ = 0, Z7= u^^z^ + 2Wj,cPj(r, -f &c. = 0, and ^ = an equation of the rfi^ order in x^^ a?„ a?g, x^ which may be written symbolically («ia;i + «,a?, + OaSCg + 042^4) =0. The method of elimination employed is to solve between the linear equations and the quadric, and substituting in ^ the two systems of values found, to multiply the results together. Now we may in an infinity of ways combine the quadric with the linear equations multiplied by arbitrary factors, so as to obtain a result resolvable into factors : that is to say, so that Z7+ (\a;j + X,a?, + &c.) [a^x^ + &c.) + [fi^x^ + &c.) [^^x^ + &c.) = [Pi^x + ^^0 (Ji^i + &c.). We shall imagine this transformation effected, but it will not be necessary to determine the actual values of X^, /ip &c. for it will be found that these quantities disappear from the result. Taking, then, the coefficient of any term xp:^ in the quadric, the equation written implies that we always have 2^,4 4 (a^X*+ a^X.) + (A>*i 4- /S^a*^) =M*+i^i?.— (^)- Instead then of solving between the quadric and the linear equations, we get the two systems of values by combining with the linear equations successively p^x^ + &c. = 0, q^x^ + &c. = 0. And by the theory of linear equations the resulting values of x^^ a?2, &c. are the determinants of the systems ?11 ?»> ?SJ ?4 ^11 «»J «8» «4 A) /^S) ^8) ^A Pil Psi Psi P4 «1J «JJ «8? «4 ^n ^2? ^3» ^4 i 328 APPLICATIO!^S OF SYMBOLICAL MEtfiODS. If then we sabstitute the first set of values in a^x^ + &c,, we get the determinant «IJ «2J «81 ^4 Piy Ps» i^8? Pa «H ^J «81 «4 which we may write [a,p,a^8,). The result of elimination may thus be written symbolically B = (ctiPfi^ffJ' (SiS^gOgiSJ"* We use in the second factor the symbol b instead of a, for the reason explained (Art. 162), in order to obtain powers of the coefficients of <l> ; but it is understood that the b symbols have exactly the same meaning as the a, since after expansion we equally replace the products a*a*a'a'^j b^b^b'b'^j by the corresponding coefficient of <f>j a^i^. We may then write the result of elimination in the more symmetrical form for tbis after expansion will be onlj double the former ex- pression. Let as now write («. P»<^A) fJ>d,<^^ + {<^AJh^,) {\PflA) = 2 C, then R may be easily expressed in terms of J, B, G. For we bare 2B = lG+^/{C*-AJB)}'+{C-^/{C'-AB)}\ or E=C'-\- " ^" ^^ C"-* ((7* - AB) 1 • iS 309. We proceed now to examine more closely the expressions for Aj B^ C, and to get rid of the quantities p and q which we have introduced, so that the result may be expressed in terms of the coefficients of the given quadric. CLEBSCH'S INVGSTiaATION OF ELIMINANT. 329 Now Ay which is the product of twd determ'mants, may be written as a single determinant, Piii i UPA+P,it), UPiq»+PA), i {Pi^4+P43i)> «ii ^u «i ^{Pii,+PA), PA , i(M.+PA)> i (i>.?4+P4?.)> «.» ^« «i i(M4+P4?.). MM4+;'4?.)> i(i'.?4+i'45'8)> P42'4 » «4. Z^^, ^4 a. /3. "4 ^4 a. 9 malttplied however by (- 1)"*^, where m Is the number of variables, that is to say, in the present case, 4. For, every element of this determinant must contain a constituent from each of the last three rows and columns ; it is therefore of the first degree in the terms p^g^^ J (Pi?»+A?i)j ^^' 5 and if the coefficient of any of these terms be examined, it will be founds according to the number of variables, to be either the same, or the same with sign changed, as in the product of the two determinants. Now in this determinant we are to substitute from equation (^), But when this change has been made, if we subtract from each of the first of the four rows and columns the a row and column each multiplied by ^\, and the fi row and column each multiplied by ^fi^j &c., the additional terms disappear, and the determinant reduces to ^111 ^W ^18J ^14J «11 ^IJ «1 ^»H ^12? ^181 ^4* ^»? ^21 ^8 ^M ^82? ^83> «*«4^ «8l ^S^ «8 ^411 «*421 «*43? ^441 «4l ^4> «4 «1 ) «2 1 «8 J «4 ^l> ^2 J ^8> ^4 «1 > ^2 ) «8 1 «4 Clebsch denotes the above determinant, in w'hich the matrix of the discriminant of a quadric is bordered by rows and uu 330 APPLICATIONS OF SYMBOLICAL METHODS. columns, a, /8, &c., by the abbreviation (see p. 18) (a, /3, a\ Va, /3, ar the upper line denoting the columns, and the lower the rows by which the matrix is bordered. Thus, then, we find for any number of variables ^ ' \a, /3, -..a/ In like manner B= (- 1)""^ («' j^' •••^) . And in the same way o=(-ir-^f«' ^' -^V ^ Va, /S, ...6/ Now it was proved, Ex. 2, p. 32, that HQ) - fjV = A h *) , where A is the discriminant of the quadric. And it is proved in the same way, in general, that /a, ^, ...a\ /a, ^, ...h\ _ fa^ ^, ...aV^ ^ /'a, ^, ..«, f>\ ^ \a, i8, ...a/ \a, ^, ...J/ \a, ^8, ...J/ Va, ^8, ...a, 6/ ' If, then, we call the last written function i), we have C*--45 = -Ai>, and the formula of last Article becomes n(n-l) n{n-l){n^2){n-S) ^n^jy^^ ^^ 1.2 ^ 1.2.3.4 310. As an example of the application of this formula we give Clebsch's investigation of the equation of the system of inflexional tangents to a cubic (Crelle^ 59), remarking in the first place that formulas of reduction corresponding to those given (Arts. 299, 300), exist for quantics in any number of variables. Thus for ternary quantics the most useful are a [bed) — b [acd) + c [abd] - d [abc) = 0, [abc] [ade) + [abd] [aec) -f [abe) [acd) = ; to which may be added the corresponding equations for con- travariant symbols (Art. 160) P[abc) = a [abc) + b [acd) + c (aai), [aab) [acd) + [abc) [aad) + [aca) [abd) =• 0, INFLEXIONAL TANGENTS TO A TERNARY CUBIC, 331 where P is a^x^ + a,aj, + a^x^. To come now to the problem proposed, the eqaatioa of the system of inflexional tangents to a cubic is got {Higher Plane Curves^ Art. 74) by eliminating between the equation of the curve and those of its first and second polars, and one of these equations being linear and another quadratic, the formula of the last article is applicable. By Art. 308 the elirainant is 4C"-3^J?(7, where A^ B^ G are respectively (JJj ^) , (^J j) , (^J J) . But we have proved, p. 18, that C3-4«(:)-K5, (::?)-8.(f)-i«ia , The result therefore is when cleared of fractions 4 LbS+6u(ff^'-S LbH+6u (jU L'H+Gu (^)| W+6«(jjl . But remembering that a' and b" are each u, this may be divided by u, and thus gives uH' + 18 PS' + 108 QHu + 216Bu', where P= 3aV (j) - «a (^) - «J (*) , ^-©'-0®®- In the above formula, as at p. 17, ^denotes the determinant formed with the unreduced second difierential coefficients ; but if we suppose that, as in the Higher Plane Curves^ each coefficient has been cleared of the numerical factor six, we must write in the above for S^ 216-H; also, since P, Q, B involve the second differential coefficients in the 2nd, 4th, 6th degrees respectively, these will be 6', 6*, 6® times the corresponding P, Q, B. Making these substitutions, the eliminant becomes uH'+ SPH' + 3 QHu H- Bu\ To reduce this further observe that a f J= — 3fi; for (j expanded is - [u^u^ - u\^) a^ + &c., but a^a (Art. 298) is u^ &c., 332 APPUCATI0N8 OF STMBOI^CAL METHODS. on making which sabstltntions, the truth of what has beefl- stated appears. In a similar manner it is easily seen that a* ( J j = — jETJ, from which at once follows that cfV ( j ) = "^ -Si*. We have then P^ SHu. In order to calculate Qj it appears from what we have just stated that •» (i) -« "» ® -»^' «■•"'■ ®® -« ® -'^- Again we can show that ah Uj = Si?"* for (Art. 33, Ex. 2) / J j =s f j K j — jyf^' , j ^ and by actnal expansion .it is easily seen that oJ [^' j) =6^?*. Collecting the terms we have ^ = - 3 J?*. Thus as the two terms dPH\ 3 QHu cancel each other, the eliminant is seen to be divisible by u and reduces to H^+Bu. 311. It remains to calculate B. Now let us in the first place observe, that if it be required to differentiate if, which is u,.. u..« u.. with regard to x^j it follows from what was said at the end of Art. 34, that the result is that is to say, the result is — a^ ( ^ J . Taking then the determinant , and multiplying the fourth ^ column by r J and the fourth row by U J , we have ^ii» ^»j ^w' ^i »n» ^M? ^28J S ^81» ^8JJ ^88) «8 hi ^2i hi ffl(°)®-©' j^j the result of substituting the differential coefficients of S for a^, a„ a, and for 6^, 6^, J^ in DOUBLE TANGENTS TO PLANE CUETES. 383 tke determmant just written. We have also (Art. 33, Ex. 2) . But if we take «i f^J and differentiate it by the same process as that already employed, we find that the second differential of H with regard to a?„ a^, is a,J, K' A , and so on. Thus it will be seen that f jj (^'jj is h (^j , by which we mean the result of substituting in (^j for a„ a,, Og symbols of differentia- tion applicable only to if. We may get another expression for (i) (a' jj* ^* ^® *^® coeflScient of X in the expansion of the Hessian of w + \ff. For, that Hessian is found by substituting in the determinant which expresses the Hessian, for each second differential coefficient Wj„ u^^^^\aJ}^ [a h)i and it is easily seen that the coefficient of X in this expansion is as stated. But if we remember [Higher Plane Curves^ Art. 225) that that ^J is what is called [Higher Plane Curves^ Art. 231), we identify the result now obtained with that there given (Art. 232, Ex. 1), viz. H'-5Su'H+uQ. 312. In the theory of double tangents to plane curves explained (Higher Plane Curves^ Art. 384), it is necessary to calculate the result of substituting in the successive emanants OgW, — a^Wj, ajW, — OjM^, a,Wj — ttjW, for x^j a?„ a?,, and to show that each result is of the form P^ + Q^ (a^aj, + a,a?, + a^>^)\ We give as a further illustration of the use of symbolical methods the application of them to the calculation of Q^y &c. These results of substitution may be symbolically expressed a*** [aau)\ a""' (aaw)', &c., where a only is a symbol. These expressions may be reduced by the help of a general formula for (aau)^, which is found as follows, as in Higher Plane Curves^ Art. 390 : 334 APPLICATIONS OF SYMBOLICAL MlETHODS. This square differs onlj in sign from the determinant ^111 ^W ^18J ^1) *n ^1 "jH ^W1 ^»3l «»1 «9» ^« ^8l» ^8J> ^837 «S1 «3> ^8 «1 ) «t » «3 • • • «I > «8 7 «8 • • • M, , M, , M, . . . But this determinant is reduced as at p. 18, so that the outside row and column become 0, 0, 0, - a, — a, — w ; and we have (-)'-»'(:)-«'(:)-^-©-«(:;:)- In the calculations we have to make, we suppose that the values of x^^ a?^, a?, make u vanish. To calculate then Q,, multiplj this equation bj a""*, the first term on the right-hand side has u for a factor and vanishes, a""* ( ) is — 3 J?", and a""* ( .) is — oH. Thus we have ^, = - a"-ff Again (?3= a"-^ (j) {aau) + a'a"^ (^) [aau) - 2a"-«a g) (aau), but the first and last terms vanish identically ; and the middle terra reduces to — a* (olHm), for we have seen that the differentials of the Hessian are - a^'^a^ ( ^ J , &c. Again, for Q^^ we have to calculate The first term vanishes, the second is 4a* f ] H^ the third is — 4a'[^j ZT, as may be seen from last Article, and since a" Q is - 35, the fourth is - 6a' (T\ H. Thus we see that the result is divisible by a', giving C.-e(>-4.(f)+«-<.^K. TYPE FORM FOR EVEN BINARY QUANTICS. 33$ 313. As final illustration of the use of symbolic methods we give typical expositions of binary quantics, in the first place, ot even degree by means of three independent quadratic covariants (Clebsch, p. 414). Let the three independent quadratics be and their Jacobians k = {Im) Ijn^y \ = (mk) mjc^^ fi = {M) kj^ and denote by accented letters symbols for the same form. Then for p = p^ any fourth quadric (compare p. 181) we have {pxY = {lm){mp){lji>)^ and writing D = - {lm){mk){kl)^ the identity Py Pi% PiPi^ Pt becomes p^D = k ( pxY + 1 (^X)' + r/i [pfif. Thus, if the required even quantic f be symbolically expressed by a^**, we find f expressed by means of. A:, Z, «i by replacing |> by a in this symbol and raising it to the power n : /D" = [{oLKyk + {a\yi + (a/A')*»*} X {[aic'yk + (a V7? + [ati'Jm] When this product is expanded the coeflScients of its terms IfVinC are evidently invariants, thus the expression is typical {p. 249)., The quadric relation which subsists between A, Z, m enables us to go back from this ternary to a binary form: introducing in so doing only the invariants of the three quadratics. If we retain A, Z, m as three variables the symbol a^ defined '^y ^y = (^^)'yi+ (^^)''y»+ (^m)V8 ^^ evidently that of a unique curve of the ri^ order, which is determined by the 2n points ofy, which are now given on a conic. The additional relations of this curve to the conic by which its uniqueness is secured arise from the fact that if the conic be written symbolically )8 ' = ^ ", then their covariant {a^^y a ""* vanishes identically* ; * F, Lindemann, Bulletin de la Soc. Math, de France, t. v. 1877. For n > 3 this epecializes the curve, thus he notices that the conditions for n = 4 limit the ternary quartic to admit of being the sum of five fourth powers, cf. Note, p. 151. 336 APPLICATIONS OP SYMBOLICAL METHODS. or again, that there is an infinity of triangles conjugate to this conic which are inscribed in the polar conic of y relative to the isurve ay* = 0; in other words, that all the conical polars of this curve are apolar to the conic ^^. 314. The exposition, however, can be made to involve invariants of lower degrees by using as its basis instead of the quadratics their Jacobians. In fact, as in Art. 193, we can see that B^^u, = i),/« + D^J^^ + D^J,, ^ u^B^ - u^B^, + u,B,,„ or, in the present notation, that k (uKy + 1 (a\y + m {afif = (oA)' k + (aTf\ + (amy /* whence /D" = {(akyK + (ary\ + {amj /i} X {{ok'yK + (aVy\ + {am'Jii] X [{ak'^jK + {airyx + (amryii). • If n be even, we can multiply out and replace all powers of /v, \, fi by functions of the same order in A;, Z, m without denominator. If n be odd, the product of all factors but one gives us even powers of ic, X, ii which can be so treated, and the last factor when expressed by A;, ?, m introduces a new factor D upon f. Thus the coefficients are the rfi^ transvectants of f on Tc^frn^^ where a + /8 + 7 = ^n, and are accordingly of much lower order than the others, which were its transvectants of the same order on k^ii^ where a + ^ + 7 = \n. It is in this manner that Clebsch developes the expression for the sextic by its quadric covariants when E does not vanish. Each covariant symbolically contained in ^y= a; [(aiyx + {amy II + (aw)V} [{aVy\ + (awOV + (anjv] is found as a linear function of Z, 9n, n by means of the invariants A^ /„ /,, ly^ and, on expanding, the products of the second order of \, /a, v are replaced by quadric functions of Z, m, n without denominator. 315. In a similar manner Dr. F. Lindemann {Maik Ann. XXIII., p. 133) has given the typical exposition of forms of the order Zp by means of four independent cubic covariants in connection with the formulaB of Art. 2195, and the consequent geometrical reference to a twisted cubic without the canonical reduction employed in Arts. 219c, d. TYPE FORM OF BINARY QUANTIC OP ORDER Bp. 3S37 In fact, DOW denoting the cubics by A/, Z/, m/, n/, and the forms -J^, Jut^-J^ «^«8 '>y '^/^ ^/^ /*x'> ^x'j it is easily seen that for any other cubic aj we have (aKy—(mn)(nr}Qm)(ar)(am)(an)j and writmg (kxy = (l\y = &c. = A, that Aa;^(aKykJ + (a\yi; + {atiym; 4- (av)V = (aA) V + (^0 V + («</^/ + («w) V- Whence we find the 3p*® y by raising this to the degree p A'aJ^ = {(a/cyk + (a\yi + (apTym + (av'/n} X [(afcyk + (a\yi + (a/A'7w + (avyn) or = {(akjie + (alJX + (atwOV + (o«')'v} X [(aky/c + (aO^X + (am'OV + (anyv) I^itber form expanded is a function of degree p in four variables with invariant coefficients. Taking the point «, = */» ^^ = ljy &c. as variable with x its locus i^ a twisted cubic, and the coordinates of the osculating planes determined by x are u^ = «„') w, = \/, &c. The equations of three quadrics through this curve or of three quadrics touched by all its osculating planes can be easily found from the obvious relation A (xxy = kjicj + Z; V + mj,ij ^riX% and the others which follow from it as successive polarsii When found these equations enable us to return to a binary system introducing only invariants of the cubics. Thus the equation of the surface, whose intersections with the twisted cubic are the 3p points of the binary quantic, is either a/ = where a, = (o/tf)'2f, + (a\)\4-(a/i)X+(av)';5^, or i8/ = where /3„ = (aA)'Wj + {al)\ + (aiw)'w, + (aw)'w^ this tangential form involving invariants of lawer order. This surface a/ = is unique, and if we denote by w/ = 0, ti<r' = 0, t«T* = 0) the three quadrics touched by all osculating planes of the twisted cubic, it is shown by Dr. Lindemann that this is because the three equations o^*a,^' = 0, ao.'a/"'=0, a^aj^^ = are identically true, and therefore a/= is conjugate, or apolar, to all the quadrics which can be inscribed in the developable surface of the twisted cubic. XX { 338 ) NOTES. mSTORT OF DETERMINANTS. (Page 1). Thb following Historicsl notioes are taken from Baltzer*s Theory of DaermtnarUs ; and from the sketch pfefized to Spottiswoode's Elementary Theorems relating to Deter- minants. The fint idea of determinants is due to Leibnitz, as Dirichlet has pointed oat. In Leibnitz's letter to L'Hdpital, 28 April, 1693 (Leibnitz's Mathematical Worht, published bj Grerbaidt, toL ii.. p. 239), is to be found the first example of the formation of these functions, and of their application to the solution of linear equa- tions ; the doable suffix notation (p. 7) is employed, and he expresses his conviction of the ferdlity of his idea. But nowhere else in his writings is there to be found any proof that he sought to-draw any new fruits from his di8(k)very ; and the method was lost until re-diBCOvered by Cramer in 1750. Cramer, in his Introduction a r Analyse des lufne* Courbes (Appendix), has exhibited the determinants arising from linear equations in the case of two and three yariables, and has indicated the law according to which they would be formed in the case of a greater number. The rule of signs by the method of displacements (p. 6) is given by Cramer. The equivalence of the other method by permutations of suffixes was afterwards proved by Bezout and liaplace. In the Histoire de VAcademie Boyale 'des Sciences, Ann6e 1764 (pub- lished in 1767), Bezout has investigated the degree of the equation resulting from the elimination of unknown quantities from a given system of equations, and has at the same time noticed several cases of determinants, without however entering upon the general law of formation, or the properties of these functions. The Histoire de VAcademie, An. 1772, part ll. (published in 1776), contains papers by Laplace and Yandermonde relating to determinants of the second, third, fourth, Ac orders. The former, in discussing a system of simultaneous di^erential equations, has given the law of formation, and shown that when two rows or columns are interchanged, the sign of the determinant is changed, and that when two are identical, the determinant vanishes. The latter employs a notation in substance identical with that which, after Mr. Sylvester, we have called the umbral notation, and explained p. 8. In his Memoir on Pyramids {Ifemoires de VAcademie de Berlin, 1773), Lagrange made an extensiye use of determinants of the third order, and demonstrated that the square of such a determinant can itself be expressed as a determinant. The next impulse to the study was given by Gauss, Disquisitiones Arkhmeticce, 1801, who showed, in the case of the second and third orders, that the product of two determinants is a deter&inant, and very completely discussed the case of determinants of the second order arising from quadratic functions, i.e. of the form i* — ac. In 1812 Binet published a memoir on this subject {Journal de VEcole Polytechnique, tome ix., cahier 16), in which he establishes the principal theorems for determinants of the second, third and fourtk orders, and applies them to geometrical problems. The next volume of the same series contains a paper, written at the same time, by Cauchy, on functions which only change sign when the variables which they contain are transposed. The second part of this paper refers immediately to determinants, and contains a large number of verj general theorems. Cauchy introduced the name " determinants," ah-eady applied by NOTES* 339 €kiT389 to the functions considered by him, and called by him " detenninants of quadratic forms." In 1826 Jacobi took possession of the new calculus, and the Tolumes of Crelle's Journal contain brilliant proofs of the power of the instrument in the hand of such a master. By his memoirs in 1841, Deformatione et proprietatibus determinarUium and De ckterminantibusfunetionalibu* (Crelle, vol. xxii.), determinants first became easily accessible to all mathematicians. Of later papers on this subject, perhaps the most important are Cay ley's papers on Skew Determinants {Crelle, vol. XXXII. and xxxvill.). Of elementary treatises on this subject, I have to mention Spottiswoode's Elemenlary Theorems relating to DeterminantSf London (1851) ; Brioschi, Za teorica dei determinanti, Favia, 1854 ; and Baltzer, Theorie und Antoen' dung der Determinantem, Leipzig, 1857 ; fifth edition, 1881. French translations both of Brio8chi*s and Baltzer's works have been published. COMMUTANTS. (Page 8). In connection with the umbral notation may b€ explained what is meant by commutantSj which are but an extension of the same idea. It is easy to see what, according to the rule of the umbral notation, is meant by £, ij, p, ^t;, t;^, , ^ & n, P, ivf n'l d d if we write for brevity f, r;, for ^ , -r . We compound the partial constituents in each column in order to find the factors in the product we want to form, and take the sum with proper signs of all possible products obtained by permuting the terms in the lower row. Thus the first example denotes ^ .ti^ — f tj . ^t;, which is the Hessian : the second denotes ^*.^'*.»?* — ^*.^»?'.^n* <fec«) which is the ordinary cubinvariant of a quartic. Again, since multiplication is performed by addition of indices, it will be readily understood that we can equally form commutants where the partial constituents are combined by addition instead of by multiplication. Thus, considering the quantics (aj, fli, floj^j yfy (a4j ««> «2> «i» OoX^i y)\ the invariants in the last two examples may be written 1,0, 2, 1, 0, which 1,0, 2,1,0, expanded are a^^ — CiC, ; a^pi^aQ — a^cLia^ + &c. All these commutants with only two rows may be written as determinants, but it is a natural extension of the above notation to form commutants with more than two rows, such as ^, ij) If 0, ^^, ^n? 1^* These all denote the sum of ^, »i, 1> 0, ^, ^, 1,2. ^, n, 1, 0, p, en, n". e, r\, 1, 0, i\ ^n, n". a number of products, each product consisting of as many factors as there are columns in the commutant and each factor being formed by compounding the constituents of the same column j and where we permute in every possible way the constituents in each row after the first. Thus the first and second examples denote the same thing, namely, the quadrinvariant of a quartic expressed in either of the forms j4,^4_4^3^. ^,,84.3^^2.^2 or a^fi^-Aa^ai + ^a^a^, while the third example {".^^ij*.!,"- Ac. denotes the cubinvariant of an octavic given at length, Art. 155. We have seen that the two invariants of a binary quartic can be expressed as commutants, but it will be found impossible to express in the same way the dis- criminant of a cubic. Thus the leading term in it being a^a^ or ^zizntnij we are naturally led to expect that it might be the commutant e, n» ^j n^ but this oommu- ^, n, ^, n, e, n, & % 310 NOTES. tant, instoadof giTing the cHscriimiumt, wOl be found to yanuh ideniicadly. It wufy howerer, be made to yield the diacriinixiant by placmg certain restrictions on the perm uta tions which are allowaUe. For further details I refer to the pi4)eEi of MeHH. Gqrley and QylTSSter in the Cambridge amd Dublin Mathematioal J<mmal, 18d2, ON RATIONAL FUNCTIONAL DBTEBIONANTS .♦ (Page 14). The determinants considered Bx. 6, 6, are partUmlar cases of the important form <> («), ^ {x) ... 4> (y), i^(y)... where ^ (a;), ^ (a;) denote rational integral functions of Zj and (y), <p {z), &Cj the same fonctions of y, e, dc respectiyely. Such a determinant may be briefly denoted by its top line | ^(x), ^{x) ... | . Thus the determinant Ex. 5 may be written I 1, Xy ^..a^^ I • This last determinant we haye seen has for its value (-l)i-(«.-i)]:*(a:,y,;5...); by which Dotation Prof. Sylyester denotes the continued prodnct of the differences (»— y) {x—z) {x^w) ... X (y— «) (y — w) ... x {z^w) Ac. This alternate product is of the nature of a square root : its square we ktoow is a symmetrical function of x^ y, e, Ac, and is unaltered by any permutation of these Tariables; but itself has two yalues corresponding to the different arrangement of the yariables, its sign being altered if we permute any two of the variables. The function 1 1, x^ ...a^> | was suggested by Cauchy as a symbolic representation of a determinant, viy. expanding it as the sum of a series of terms ±y^zhtfl ..., and changing the exponents into suffixes, or the term into x^iZ^to^ ..., we have the development of the determinant x^ Xi,.,xn-x It may be further remarked in passing, that any rational function of the vaiiahlei X, y, &c,f which, however the variables are permuted, has only two values, most be of the form P+ Q$^, where P, Q are symmetric functions of the variables. Returning now to the general determinant | <i>{x), \J/{x).,. | , it obviously con" tains ^i as a factor, for on the supposition x=y, it vanishes as having two rows the same, and is therefore divisible by x— y; and similarly with regard to every other difference. Let us theji in particular examine | ar*, x^, x^... | , which we may caD I a, /?, y, ... I in order to ^nd the value of the remaining factor. If a be the least of these exponents, we may divide each row by a*, y*, ... respectively, so that we can at once reduce the investigation to that of the case where o = 0. In the following we employ a method given by Jacobi, De ^unetionibus AlUr- fMfUibuSf CreUe 22, (1841); depending on the consideration of the determiiuoit 1 1 1 For convenience we work with the case of three x—a*x — b*x — e"' * variables, but it will be seen that the process is perfectly general. Consider then the equation which is obviously true ^ l^ J_ {B—a* X— 6' x—c 111 y-a' y-6* y-e 111 2-a* ff-b^ z-c ^{xjy,z)l^{a,b,e) (x-a)(x-6)(x-c)(y-a) {3/-b){y-c) («-a)(a-^) («-*)' ♦ This note is, in substance, Professor Cayle3r*8. NOTES. 341 «ul expand tach-side bj ddoending powen of x, y, e. We have 1 1 . a . a^ . J, «— a X ar XT whence, on the left-hand side, the term multiplying the reciprocal of x'y^z^ la a*-S 6*-S c«-» a/3-S *^-S c^-i avi, 6T-1, c-r-i In order to expand the right-hand side, observe first that if Hu H^ ^. have the same meaning as at Art. 272, then 1 _ 1 ^1 . ^ .^^ (« — a) {x — b) (x— c) ~ a* ar* a:* *' as is easy to see by multiplying together the expansions for dec. We haye also S* («> y> «) = - f i> »» ** i> yi y^ 1, «, «2 Hence, the right-hand side is ^ (a, i, c) multiplied by i + §^4c.. ^ + 1 + 40.. 1 + 5+40. and the term multiplying the reciprocal of x^yi^z^ is ^' (a, 6, «) multiplied by Sa-%y ^a-2f -"o"! ■^/3-» -SJ3_a, J^_i We haye thus e^», *«-», <?^i o^-i, d^-\ c/3-1 ■"y-»» ^y-t) ^y-\ = ^{fl,b,c) 'a-w Sp-t> fffi-^ B^-i •2^-81 "^v-a* -^v-i which we may write (a-1, ^-1, y-l) = r (a, J, c) Jr(o-3, ^-8, y-3). We may verify this equation by writing a = l, /3=2, y = 3, observing that ji (a, 6, c) = — (0, 1, 2), that H_^ H.^ vanish and that iTo is 1* If we write a = 1, and for /3, y write /3+1, y + 1, we have (0,Ay) = r^(-2,^-2,y-2) = r* y-J, "y-u -"-y But since iLj, 5-i vanish, and H^ = 1, the last determinant reduces to Thus we have finally (0, ^, y) = $' (-Hjs-a^y-i -5j3-ii?y-2). As an example, taking /5 = 1, we get (0, 1, y) =— iry_j$ , a formula which includes that of Ex. 6, p. 16. We may also consider determinants involving the square roots of rational functions J{</»(a;)}, x^[ylf{x)], ... '\{<t> (y)}, y W (y)}» - but these, although presenting themselves in the theories of Elliptic and Abelian functions, have been but little studied. HESSIAN. (Page 17). The name was given by Sylvester after Professor Otto Hesse, who made much use of the functions in question, which he called functional determinants. They are a particular case of those studied under the same -name by Jacobi {CrtUe 342 NOTES. ToL XXII.)) the oonatitaents of which are the differentials of a series of n homo* geneous fonctioxis in n Tariables. It is so convenient to have short distinctive names for the functions of which we have repeatedly occasion to speak, that I have followed Sylvester in calling the former Hessians, the latter Jacobians, see Art. 88. SYMMETRIC FUNCTIONS. (Page 66). The rules for the weight and order of symmetric functions are Prof. Cayley's. The formula, Art 59, I have taken from ;Serret'8 Lessons on ffightr Algebra, The differential equation. Art. 60, is an anticipation of the differential equation for invariants, of which I speak. Art. 143. Brioschi (see M. Roberts, Q^arterly Journal^ yoL IV. p. 168), remarked that the operation {ydx} (Art. 65), expressed in terms of the ELIMINATION. (Page 66). The name ^eliminant' was introduced I think by Professor De Morgan ; I believe I have done wrong in using a second appellation when a name to which there was no objection was already in use. The older nazne ' resultant ' was employed by Bezout^ Histoire de VAeademie de Paris, 1764. The method of elimination by symmetric functions is due to Euler {Berlin Memoirs, 1748). The reduction of the resultant to that of a linear system was made simultaneously by Euler {Berlin Memoirs, 1764) and Bezout {Paris Memoirs, 1764). The theorem as to the degree of the resultant is Bezont's. The method used in Art. 74 of forming symmetric functions of the common values of a system of two or more equations is Poisson's (see Journal de VEcole Polytechnique, Cahier xi.). Sylvester gave his mode of elimination in the Philosophical Magazine for 1840, and called it ^ dialytical,' because the process as it were dissolves the relationa which connect the different combinations of powers of the variables and treats them as simple independent quantities. Cayle/s statement of Bezout's method is to be found, Crelle, vol. Liii., p. 366. Sylvester's results in Art. 91 are to be found in the Cambridge and Dublin Mathematical Journal for 1852, vol. vii., p. 68 j and Cayley's general theory (Art. 92, Ac.) in the same Journal, vol. in., p. 116. It was noticed by Lagrange, that when two equations have two sets of common roots, the differential of the resultant with respect to the last term vanishes (see Berlin Memoirs, 1770). Sylvester showed, in January, 1858, that the same was true of all the differentials, Cambindge and Dublin Mathematical Journal, vol. viii., p. 64. He showed at the same time, that the common roots were given by the ratios of the differentials. The proof in Art. 99 is, 1 believe, my own. The theorem. Art. 99, is Jacobi's Crelle, voL xv., p. 106. In this part I have made some use of the Treatise on Elimination by JB^a4 de Bruno. The theorem of Art. 102 is Prof. Cayley's. DISCRIMINANTS. (Page 98). The word * discriminant ' was introduced by Sylvester in 1852, Cambridge and Dublin Mathematical Journal, vol. vi., p. 52. The word 'determinant* had been previously used, and had come to have a perplexing variety of significations. The theorem referred to, Not^, Art. HI, was the basis of my investigations {Cambridge and Dublin Mathematical Journal, 1847 and 1849) on the nature of cones circumscribing surfaces having multiple lines. If the equation of a surf ace hebQ + b^x + b^+ix^ and if a^ be a double line, y must be contained by b^ in the second and bi in the fixat degree. The discriminant with respect to x is a tangent cone which has g* for a factoi. NOTES. 343 BEZOUTIANTS. (Page 107). It bad been abown (Art. 85) that the resultant of two equations of the n** degree is expressed by Bezout's method as a symmetrical determinant. This may be considered (Art. 118) as the discriminant of a quadratic function which Sylvester has called the Bezoutiant of the system. When the quantics are the two differentials of the same quantic, th^i if we resolve the Bezoutiant into a sum of squares (Art. 165), the number of negative squares in this sum will indicate the number of pairs of imaginary roots in the quantic. The number of negative squares is found by adding (as in Art. 46) \ to each of the terms in the leading diagonal of the matrix of the Bezoutiant, and then determining by Des Cartes' rule the number of negative roots in the equation for X. The result of this method is to substitute for the leading terms in Sturm's functions, terms which are symmetrical with respect to both ends of the quantic ; that is to say, which do not alter when for x we substitute its reciprocal (see Sylvester's Memoir, Philosophical Ti^ansactions, 1853, p. 513). LINEAR TRANSFORMATIONS. (Page 108). The germ of the principle of invariance may be traced to Lagrange, who, in the Berlin Memoirs, 1773, p. 265, established the invariance of the discriminant of the quadratic form ax'^+'ibxy+cy\ when for x is substituted x + \y. Gauss, in his Bisquisitiones ArithmeticoB (1801), investigate^ very completely the theory of the general linear transformation as applied to lunary and ternary quadratic fcMrms, and, in par- ticular, established the invariance of their discriminants. This property of invariance was shown to belong to discriminants generally by the late Professor Boole, who, in a remarkable paper, Cambridge Mathematical Journal, 1841, vol. ill., pp. 1, 106, applied it to the theory of orthogonal substitutions. He there showed how to form simultaneous invariants of a system of two functions of the same degree by performing on the d d discriminant of one of them the operation a' :r + ^' jI + ^^' Boole's paper led to da ao C^yley's proposing to himself the problem to determine a priori what functions of the coefficients of an equation possess this property of invariance. He found that it was not peculiar to discriminants, and he discovered other functions of the co- efficients of an equation at first called by him * hyper-determinants,' possessing the same property. Cay ley's first results were published in 1845 (Cambridge Mathe- maticalJournal, vol iv., p 193). From this discovery of Cayley's, the modem algebra which forms the subject of the bulk of this volume may be said to take its rise. Among the first invariants distinct from discriminants, which were thus brought to light, were the quadrinvariants of binary quantics, and in particular the invariant 8 of a quartic. Mr. Boole next discovered the other invariant 7" of a quartic, and the expression of the discriminant in terms of S and T {Cambridge Mathematical Journal, vol. IV., p. 208). It is worthy of notice that both the functions S and T had been used by Eisenstein {Crelle, 1844, xxvii , p. 81) in his expression for the general solution of a quartic, but their property of invariance was unknown to him, as well as the expression for the discriminant in terms of them. Cayley next (1846) published the symbolical method of finding invariants, explained in Lesson Xiv. {Cambridge and Dublin Mathematical Journal, vol. I., p. 104, Crelle, vol. xxx.). The next important paper was by Aronhold, 1849 {Crelle, vol. xxxix., p. 140), in which the existence of the invariants 8 and T' of a ternary cubic was demonstrated. Early in 1851 Mr. Boole reproduced, with additions, his paper on Linear Transformations {Cambridge and Dublin Mathematical Journal, vol. vi., p. 87), and Sylvester began his series of papers in the same Journal on the Calculus of Forms, after which discoveries followed 344 NOTES. ia tapid saccession. I can scarcely pretend to be able to aseign to tbeir ptogtif aathoTS the merits of the seyend steps; and, as between ^Klessra Cayley audi Sylvester, perhaps these gentlemen themselves, who were in constant communication with each other at the time, would now find it hard to say how mndh properly belongs to each. To Hr. Boole {Cambridge cmd Dublin Mathematical Journal, ToL YI., p. 95, January, 1851) is, I belieye, dne the principle that in a binary quantic the' (q;)eratiTe symbols -j-, — -j- voslj be snbstitated for x and y. The prindide was extended to qoantics in general by Sylvester, to yritiom is to be ascribed the general statement of the theory of contravariants, Cambridge and Dublin Mathemaiieal Journal, (1857), voL YI., p. 291 ; although particular applications of oDntravariants had pre- viously been made in Geometry in the theory of Polar Reciprocals, and in the theoiy €^ ternary quadratic forms by Gauss {Disquisiiiones AnthmeUecB, Art. 267), who gives the reciprocal under the name of the adjunctive form, and estaUishes its inrariance under what he calls the " transformed substitution.'' Sylvester also re« marked thart we might not only replace contravariant by operativo symbols, but also du du by the actual dififerentiab ^» 7~) ^<^* ^o Boole I would ascribe the principle (Art. 126) that invariants of emsnants are oovariants of the quantic (1842), Cambridge Mathematical Journal, vol. III., p. 110, though BooWs methods w^re generalized by Sylvester, Cambridge and Dublin Mathematical JowrnaX, vol. vi., p. 190. Some of the first steps in the general theory of covariants may thus be ascribed to Boole, though a remarkable use of such a function had been made by Hesse in determining the points of inflexion of plane curves. I had myiself been led to study the same functions both for curves and surfaces, in ignorance of what Hesse had done {CamXfridge and Dublin Mathematical Journal, vol. ll., p. 74). The discovery of evectants (Art. 134) is Hermite's, Cambridge and Dublin Mathematical Journal, vol. vi., p. 292. In Cayley*s £rst paper he gave a system of partial differential equations satisfied by invariants of functions linear in any number of sets of variables. The partial differential equations (Art. 149) satisfied by the invariants and covariants of Innary quantics were, as far as I know, first g^ven in print by Sylvester {Cambridge and Dublin Mathematical Journal, vol. vii., p. 211). Sylvester there acknowledges himself to have been indebted to an idea communicated to him in conversation bj Cayley ; and he also speaks of having heard it said that Aronhold also was in pos- session of a system of differential equations. These are not made use of in Aronhold's paper (jCrelle, vol. xxxix.) already referred to, but he refers, Crelle, vol. lxil, to a communication made by him in 1851 to the Philosophical Faculty at Konigsberg, which, if it ever appeared in print, I have not seen. Very probably there may be other parts of the theory to which Aronhold may justly lay claim. After the publication in Crelle, vol. xxx., of Cayley's paper, in which the symbolical method of forming in- variants was fully explained, Aronhold worked at the theory in Germany simultaneously with the labours of Cayley and Sylvester in England ; and the mastery of the subject exhibited by his papers leads me to suppose that of some of the principles he mnit be able to claim independent if not prior discovery. The method in which the subject is introduced (Art. 121) is taken from his paper {Crelle, vol. LXii). I refer in a note on next page to the valuable paper by Hermite {Cambridge and Dublin Mathematieal Journal, vol. ix., p. 172), in which the theorem of reciprocity was established, whidi had at first suggested itself to Sylvester, but was hastily rejected by him, and is which the whole theory of quintics received important additions. Mixed con- comitants are Sylvester's {Cambridge and Dublin Mathematical Journal, vol. vii p. 80). The theorem, Art. 135, is Cayley's and Sylvester's. The application of sym- metric functions to the invariants of binary quantics was, I believe, first made in the Appendix to my Higher Plane Curves (1852) . The method (Art. 138) of thence finding KOTEd. 345 ooaditions lor systems of equalities between the roots is Cayley^s {Philosophical Trfmsactiom, 1857, p. 708). With regard to the subject generally, reference must be made to t^ia important series of papers by Sylvester, beginning in the sixth Tolnme of the Camhridge and Dttblin Mathematical Journal ; to a series of papers on Quantics published by Cayley in the Philosophical Transactions ; and to Aronhold's Memoir on Inyariants {Crelle, vol. LXii.). The name 'inyariant,' as well as much of the rest of the nomenclature, is Sylvester's. CANONICAL FORMS. (Page 160). The name is Hermite's ; the theory explained in this Lesson is Sylvester's, see a paper [Philosophical Magazine, November, 1851) published separately, with a sup- plement, in the same year, with the title An Essay on Canonical Forms, COMBINANTS. (Page. 161). The theory of combinants is Sylvester's, Cambridge and Dublin Mathematical Journal (1853), vol. viii., p. 63. In the case of the resultant of two equations it had, I think, been previously shown by Jacobi, that the resultant of \u+iiVy \'u + fi'v was the resultant of «, v multiplied by a power of {\fi' — \'fi). Sylvester's results Arts. 185, 188, 189, are given in the Comptes rendus, vol. Lviii., p. 1074—9. APPLICATIONS TO BINARY QUANTICS. (Page 176). In Lesson zvii the discussion of the quadratic, cubic, and quartic, is mainly Prof. Cay ley's. See his Memoirs on Quantics in the Philosophical Transactions, 1854 The second form of the resultant of two quadratics, p. 180, is, as elsewhere stated, Dr. Boole's. Sylvester proved {Philosophical Magazine, April, 1853) that every invariant of a quartic is a rational function of S and T. The theorem. Art. 206, that the quartic may be reduced to its canonical form by real substitutions, is Legendre's {Traits des Fonctions Elliptiques, chap. ii). The discussion of the systems of quadratic and cubic, two cubics, and t^ro quartics, was, I believe, for the most part new, when it appeared in the second edition in 1866. The form for the resultant of two cubics, obtained by him by a different method, was published by Clebsch {Crelle, vol. LXiv.), but had been previously in my possession by the method giyen in Art. 213. On the connection p. 174 between concomitants of binary systems and those of a larger number of variables, R. Sturm's paper (Borchardt, lxxxvi. pp. 116-46) should bo referred to. Also for the reduction of the system of two quartics, p. 224, announced by Sylvester, see Stroh, Math. Ann. xxii. 293, who cites d'Ovidio as having also effected it. In Lesson xviii the canonical form of the quintic aafi + hy^ +cz'^, which so much facilitates its discussion, was given by Sylvester in his Essay on Canonical Forms, 1851. The invariants J and K were calculated by Prof. Cayley. The value of the discriminant and its resolution into the sum of products (p. 230) was given by me in 1850 {Cambridge and Dublin Mathematical Journal, vol. v. p. 154). Some most important steps in the theory of the quintic were made in Hermite's paper in the Cambridge and DtUflin Mathematical Journal, 1854, vol. ix., p. 172, where the number of independent invariants was established ; the invariant 1 was discovered ; attention was called to the linear covariants -, and the possibility demonstrated of expressing by invariants the conditions of the reality of the roots of all equations of odd degrees. The theory of the quintic was further advanced by Sylvester's "Trilogy** {Philo- sophical Transactions, 1864, p. 679) ; and in Hermite's series of papers in the first volume of the Comptes rendus for 1866 already referred to. The values of the iiiV£uiants A, B, C of the sextic were given by Prof. Cayley in his papers on Quantics, Y Y 846 NOTES. and the existence of the inyariant E pointed out. The rest of what is stat^ In the text about the sextic in Arts 252-6, 260-1 is nearly as it appeared for the first time in the second edition. Arts. 267-9 are from Clebsch *< Theories <fec.," p. 297, and Arts. 262 to the end of the Lesson are from the sonroes indicated in the foot-notes. The term ''apolar" is dae to Th. Beye, whose investigations on << Moments of Inertia, ^.j" Borchardt, Journal lxxii, led to his '^ Erweiterung der Polarentheorie algebraischer Flachen," vol. Lxxvm, p. 97, which he opens by remarking, that the polar theory of snrf aces of the n** order has hitherto dealt only with polars of points taken singly or in groups. But in regard to any such surface Fn there also corresponds to any surface of class Ar, for A; < n, a definite surface of order n — k which become^ identical with the polar of a group of k points in case the surface of class k reduces to such a group. He claims a special interest for such surfaces of class k for which every arlntrary surface of order n— k may be regarded as polar in regard to the surface /V, and calls them ^' apolar to the surface F«," because to them no definite polar belongs. Bosanes had previously (Borchardt, lxxyi, p. . 313) termed two binary forms of the same degree whose quadriuvariant vanishes ^* conjugate to each other.** To explain the relation between the terms and connect them with the process of transvection, let Xj, x^ ... be any variables, and t<„ u^ ... contragredient to them, so that when both are linearly transformed, the foi-mer by a direct and the latter by its reciprocal substitution the value of u^Xi + u^x^ + u^z + ^ unchanged, then any relation among the former may be expressed symbolically by the vanishing of an expression (a,x, + OjOj, + ajO?, +)" = a«", and any relation among the latter by that of (o,tt, + o^Kij + a,«8 +)p = aj^. We may refer to these as the locus Ox" and the envelope aj'. Substituting for u^ ttj, .. -^, -j-, ... in Ot^y supposing p <n, and operating with the result on rtx" a new locus is found, ac^ax*'^ = aa^ax*~Pi which is called a polar of oi^ in respect of a.x*j it being obvious that x>olars of higher degrees may be formed by repeating the operation, if need be, on additional factors of the symbol Ox". Again, supposing^ > », and substituting for a;,, arj? ••• j~ > T~ » in Oa;", and operating on a«P, we find a new envelope aa^c^J^^ = aa^at?~^, which should be similarly considered a pole of Ox" in respect of a„P. When n—py the derived function is an invariant whose vanishing Oa" = oti^ — ^ is the condition for aj^ = 0, and o«" = to be "conjugate." If the form aj^au^~^ or aa"«tt'^" vanish for all values of the variables «, a*" ip "apolar" to a^J^^ and if Cx^ be any other function such that aj^bj^^ = cj^^ then aJP = 0, or an** is conjugate to any form which contains ax** as a factor. TABLE OF A FEW TBANSVECTANTS OF A QUANTIC OF ANY DEGBEE. Let the quantic of degree v be written with binomial coe^cients i/j, v^, dec, a — afficv + Via^x^~^y + y^a^'^y^ +, and let a covariant b whose source d^ is a function of a^j aj, ... supposed cleared of any common numerical factor, so that it is in its lowest terms, be of the degree f>, then with binomial ooeflftcients b = b^^P + pib^xp'^y +. The source may be named without su^, thus, for the Hessian h = o^a, — a,', k = a^a^ — ba^aia^ + ^a^^a^^ — Qa^a^ + %a^a^y f = a^a^ — 6aia^ + 150,04 — lO^j't I = OoOjOe ~ 3a^a,flrj + 2a<,a,? - a^ai^ + SaiO^^ — a^a^a^ — Za^a^ + 2ajO,', o = a^a^ — 7ao«i«6 + 9«o^^5 — ^^qCz^^a + 200,03' — 30o,O2«4 + 12a,*04, » p — a^a^ — 80,07 + 280205 — 5603O5 + 3504', q = 200^6 (^o*'^* — 40,03 + 302^) - Soo'cfs' + 18000,0405 — 12o<,a2a3a6 — ^la-^a^ - l^a^a^a^ + 1080,0^0304 - 54o2%4 + I600O3204 - 64o,03» + 36022032 = 12(toi2- >V)> TABLES. 'Ml e ^ (ao%3 — 3000,02 + 2aj») O5 — 00^04* + 3090,0304 + 4oo02*04 - 5o,%204 — ia^a^^^ - 2o,«o,« + 8aiO,«o, - 3o,* j also let aj^i — afi^ = (o^),) Oq^j " ^1^1 + ^9 = {'^)tt ^^'t ^^^ ^^ transvectant 2 (oA), = ^ ; 2 (2i; - 5) (oA)2 = (1/ - 3) io ; 4 (2i; - 5) (oA), = (1; - 4) * j 4 (2i; - 6) (2ir - 7) (oA)4 = (v - 4) (v - &)/o + 12 (1; - 1) (2ir - 7)y J 8 (2v - 6) {2v - 7) (oA)5 = (v - 5) (1/ - 6) 0; 8(2v-5)(2i/-7)(2i/-9)(oA).=(v-6)(i;-6)(i/-7)i)o + 10(v-l)(2v-9)(5v-22)/; (fee. 2 (ot), = At; 2 (2i/ - 9) (oi)^ = ^v - 5)/o - 6 (2v - 9)/ ; 4 (2ir - 9) (at), = (v - 6) o ; 4 (2i; - 9) (2v - 11) (o»)4 = (v - 6)(v-7)i>o -4 (v - 12) (2if - 11) /; <fec.,' 2 {hi) 1 = a^cLfig-aQa^a^ — a^a^^ — 2a^a^a^^ + Sai*a^ + ^.a^a^d^ — a^a^a^ — Coj^OjO, + 3ojOj» = 3 (q^')i ; 2(2v- 5) (2ir - 9) (Ai)2 = (^-5) (2ir - 5) Zo - (2v - 7) i^^ (2v - 5) (if- 6) c j 3 (3v- 13) (o;), = (v - 6) to + (3v - 13) e; Ac. The additional functioii of a^^ a, dec. appeais on p. 237 as (9), where also e is (8) with altered sign^ or — Jf of Art. 235. The other boidx»b A, g^ t, h^f^ 0, p are the series giTe^ in the end of Art. 192 \ j is equally well known, and Z and q begin with the sexticy Arts. 252, 257. The 23 forms of the complbts system of thh quintic a may now be described as follows: From o determine its covariants A, t, /, e, form also the linear oovariant a = {ij\ by operating with ionj; the quadrico variant t of the cabic j ; and the Jaoobian -d* of the qnadrics i, t. We have then As invariarUs: the invariant of t, of i and r, of t, and the eliminant of 2r, a. As linear covariants : a, and its Jacobians with », x, and 3*. The quadric covariants : », t, d*. The cubics : J and its Jacobians with i and t. The quartics: e and its Jacobian with ». The quintics: a and its Jacobiana with if {k) and with x. The sextics: h and its Jacobian with i. The septic: the Jaoobian of A with^ ; and the nonic: the Jacobian of o with A, (^). The 26 forms of the gompletb system of the sextig a may also be thus described: From a determine its covariants A, i,/, l, q ; form also the qnadric covariants m, n (Art. 259), which are {li)^, (mt)2. We have then The invariants : /, {aj )«, the invariant of /, that of m, and (Zmn}. The quadrics : /, m, n and their Jacobians \, fi, v. The quartics : », g, and the Jacobians of i with /, and of q with / and with m. The sextics: a^j, and the Jacobians of a with Z and with m, and ofy with /. The octavics : A, kj and the Jacobian of A with /. The decimic : the Jaoobian of h with i 'f and the duodedmic : the Jacobian of o with A, (^). In this notation we have the following sums of poweks of differences OF the roots by help of which the first few terms in the equation for the squares of the differences can be calculated. They were given by M. Eoberts {Quarterly Journal, vol. iv. p. 173), in whose papers will be found several interesting' relations among the covariants of binary quantics. a*I, (o - ^)2 = - 1/2 (1/ - 1) A, a*L (o - /?)* = v^{v- 1) {v^h^-i (1/ - 2) (w - 3) aH}^ o«X (o ~ ^)« = v^{v- 1) {- p^k\+ iv^ {u - 2) (1/ - 5) o^At - ii/ (1; - 2) (7i/ - 15) o^ - 12^5 (" -2){v- 3) {u-- 4) {u - 5) aYl> 348 TABLES. a«2:(a-/3)« = p»(i/-l) W-ii^(i'-2)(v-7)a2A*»-2y»(v-2)(3v-7)o»Ai + ,Vy« (ir -2) (ir-8) (i/* +81. -21)a*t« + ^i/* (v - 2) (v - 3) (1;- 4) (ir - 21)tf«V' TABLE OP EXPANDED EESULTANTS, CHIEFLY OF EQUATIONS WRITTEN WITHOUT BINOMIAL COEFFICIENTS. 1. The resultant of the two quadratics {A, B, Cj[xj yf, (o, 6, ej^, yf is {Ac - Cay- {Ab - Ba) (Be - Cb), or a^C^-'abBC+ae{B*-2AC)+b^AC-bcAB+(^A*, 2. The resultant of the quadratic (^4, B, c\x^ y)« and the cubic (a, 5, c, d]5[«, y,)», is a«(7» - aftJ?(7« + acC{Ifl- 2AC) -ad{B»- SABC) + b^AC* - bcABC+hdA (B«- 2AC) + d^A^C- cdBA^ + €^AK 8. The resultant of quadratic and quartic is o«e» - abBC» + acC* {B' - 2AC) - adC {B* " ZABC) + ae {B*-^kB*AC+ 2A^C^ + iPAC* - bcABC^ + M^C7(B« - 2AC) -b€A(B*- ZABC) + (?A*C^ - cdAWC + ccii« (J?* - 2i4(7) + cP-4»C- <fe^»5 + ^^*. 4. The resultant of quadratic and quintic is a^O - obBC* + acC^ {B^ - 2AC) - adC^ (5» - 3ABC) + aeC{B*- ^LAB^C + lA^O) - af {B^ - hB^AC + bA^BC^ + b'AC*- bcABC* + bdAC^ {B^ - 2AC) - beAC {B^ - SABC) + b/A {B< - iAB^C+2A^C^ + c^A^C^ - cdA^BC^ + ceA^C {B^-2AC)- c/A^ (£» - 3ABC) + d^A^C^ - deA»BC+ d/A' (B^ - 2AC) + e^A*C-efBA* +f^AK 5. J)iscriminant of cubic is 27 AW^ + ^AC^ + 4DB* - mC^ - ISABCD. 6. Resultant of two cubics (Aj B, C. Dj[xf y)*, (a, b, c, d^J^a;, y)*. The value expressed in terms of the determinants of the form Ab — Baia given in p. 77 (and for forms with binomial coefficients, p. 207). Expanded it ia a^I>^ - a%Ciy^ + a^cB {C^ - 2BD) - a^d (C^ - SBCB + 3AB^) + ab^BD^ - abcD {BC - SAD) + abd (BC^ - 2B^D - ACB) + ax^B {B^ -2 AC) + acd {2AC^ + ABB - B^C) + ad^ [B^ - 3 ABC + SAW) - b^AD^ + I/^cACD - b'^dA (C* - 2BD) - bc^ABB + bcdA {BC- 3AB) - bd^A [B^ - 2AC) + c^AW - c'^dA^C + cd:^A^B - d^AK 7. The discriminant of a quartic written with binomial coefficients, expanded is aV - 12a^bde^ - ISaVe^ + bia^cd^e - 27aH'^ + h^ab^c^^ - Sab^d'e - 180aic*<fc + lOBabcd^ + 81ac*e - biac^d^ - 27b*e^ + lOSb^cde - G-kb^d^ - 546Ve + 364W. RESULTANTS. 349 8. The diflcriminant of a quartic written without binomial coefficients is 4 {I2ae -Qbd+ c^)* - {72ace + 9bcd - 27acP - 27c*« - 2(?»)«, or exx>anding and dividing by 27, 266o3e» - 192<^bde^ - 128oVe2 + lUa^ccPe - 27a'^d^ + 144aJ«c6« - Bab^cPe - SOabc^de + 18a6c<f» + 16ac*e - iac^d^ - 276*6* + isb»cde - 46»d:» - 46Vtf + 6V(P. 9. The resultant of cuHc {A, B, C, d\x^ yf and quartic (a, 6, c, ^, e!^, y)* is a»Z)* - aViCD^ + a«cl>2 (C* - 2BD) - a^dD (C* - dBCB + dAJ>^ + aH (C* - ^C^B + 2B22>2 + 4ACB^ + rtft^-BD* - abcB^ {BC - 8^1D) + aMD {BC^ - 2522) - ACD) - abe {BC^ - 3B^CB - AC^B + hABBP) + a^BI^ (B^ -2AC) + ocdB {2AC^ + ABB - B^C) - + ace {B^C^ - 2AC* - 2BB* + ^BCB - dA^B^) + ad^B {B^ - ZABC+ BA^D) - ade {B^C- BABC^ - AB^D + HA'CB) + ae2 {B* - A^AB^C + 242^2 + 4J25i)) - h^AB^ + hi^ACBI^ - i2<?.4D (C^ - 2B2)) + hHA (C7» - 3B(7i> + BAjy^) - hd^ABB* + JcrfA2> {BC- BAB) + 6cc^ (2^i) + ACB - BC^ - bd^AB {B^ - 2.167 + bdeA {B'C - 2^e2 - ABB) - b(flA {B* - BABC + 3^2j)) + c«^2i>2 _ cHA^CB + c2eJ2 (C7» - 25i)) + cd^AWB - cdeX* (5(7 - 8AD) +;ce2^2 (52 - 2^(7) - <P-4»i) + d^eA*C - (fe2^»5 + ^A\ 10. The resultant of two qnartics (-4, 5, C, D, ^]|^a;, y)*, (a, 6, c, d, e^^a;, y)* is (cf. pp. 220, 277), a*E* - a*bBE* + a^cE' {B^ - 2CE) - cfldB (Z)» - BCBJS + BBJE^ + aH {B* - ^CD^E + 2C^E^ + 4£BE^ - 4AE^) + a^b'^CE^ - a^bcE^ {CB - 35^ + a^bdE {CB^ -2C^E- BBE + 4AE^ - a^be {CB» - SC^D-E - BB<^E + 55C^ + ^Di;*) + a^c^E^ (C7» - 25i)) - a^cdE {CW - 252)2 _ bCE + 5^5E) + a^ce {C^B^ - 255» - 2C^E + 45C2)JE; + 2.1525 - 35252 + 2ACE^ + a^d'E (e» - 35C2) + 8^52 + 35*5 - BACE) - a'^de (C*B - BBCB^ + BAB^ - 5C725 + 65255 - 2ACBE - 5.4552) + o2e2 (C* - 45(725 + 25252 + AACB^ + 4.B^CE - 2AC^E - 9ABBE + U?E^ - ab^BE»^ + ah'^cE^ {BB - 4.45) - ab^dE {BB^ - 2BCE - ABE) + ab'^e {BB* - 35005 - .45*5 + 35252 + 2.4 CE^) - abd^E^ {BC - 3.45) + abcdE {BCD - BAD^ - 3525 + 4.4 C5) - abce {BCB^ - BAB^ - 25^25 - 5255 + 8.4(755 - 2.4552) - abd^E {BC* - 2525 - ACD + 5.455) + ahde {BC^B - 25252 - ACD^ - B^CE + lOABBE - 8.4252) - o*e2 (508 -^ 352(75 - .4(725 + 5J552 + 35»5 - 2.45(75 - 5.4255) + ac«52 (52 - 2.4(7) - ac2d5 (525 - 2.4(75 - A65) + ac2e (5252 - TLACD^ - 252(75 + 4.4(725 - 4.4252) + acd^E {B^C - 2^(72 - ABB + 4.4*5) 350 TABLES. - acde {B*CJ) - 2AC*D - ABU» - ZE^E + %ABCE - 2A^DE) + «c«« (B»C« - 2iiC» - 2B»i) + 4il5C/> - 3ii«l>« + 2i4B2J5; + 2A^CE) - a<P^ (^ - ZABC + 3^«i)) + «Pe (B»i) - 3^5C7i) + 3.4«2>« - AB^E + 2^«CjE:) - afe« (B»C - 3.4B(7« - AB^D + 6.4«Ci) + ^«5^ + a«» (^ - -LIB^C + 2^«C7« + 4^«^i) - -Ll*^ + d*^^ - IficAJDE* + ft»d:A£ (1>« - 2CE) - 6»<4 (i)« - ^CBE + 3BiP) + 6V^C£;*-6«tf«Li^ {CD -ZEE) + 6Ve^ (aZ>« - 2(7«^ - BEE + 4^^) + h^cPAE (C7« - 25i) + 2.4^0 ^b*deA(C*E-2BE'-BCE+5AEE) + bh^A{C*-ZBCE + BAjD^+SE^E-dACE) - ftc»-4^-S« + bd^dAE {BE - LAE) - Ix^eA {BE* - 2BC:E - AEE) - 6c<P-4^ (^C - BAE) + fctfe^ (^CD - BAE* - 3^£ + 4^C£) - bce^A {BC* - 2B*E -ACE + hABE) + bd^AE {B^ - 2AC) - M*e^ (B»i) - 2ACE - ABE) + bde'*A{B*C' 2AC* - ABE + ^A*E) - b^A (B» - 3 ABC + SAW) + c^A^E* - c^dA^EE + c»<l4« (i>» - iCEj + f^d^A^CE - <NkA* {CE - BEE) + c«e«^« (C* - 2^2)) - cd^A^BE + cd^eA* {BE - 4J^ - c<fe«^« (^C7 - 3AZ>) + e^A^ (^ - 2ul(?) + d^A*E - rfV^*/) + d'e*A*C - d^A^B + c*^*. TABLES OP SYMMETRIC FUNCTIONS as calculated by Meyer Hirsch and yerified by Prof. Cayley. They have been since extended to the eleventh degree by M. Fall de Bruno (see his Theorie de» Formes Binairesjj and in the American Journal of Mathematics to the twdfth degree by Mr. W. P. Durfee, voL v. p. 46 and p. 348, and to the thirteenth by Capt. P. A. MacMahon, vol. vi. p. 289. The equation is supposed to be a:"+6a?»-*+«c*"*+dfc.=0. I. 2a = - b, IL Xa2 = 62-2c;2a^ = c. HL Xa» = - 6» + 36c - 3rfj Xa^/? = - 5c> 3rfj 2a/3y = - i IV. Za* - 6*-462c + 2c2 + 4W-4€j Xa3^ = 62<j-2c2-W + 4tf. Za*^ = c*-2irf + 2€j So«^y = 6(?-4e; 2a/3yd=:«. V. So* =-6* + 568c-66c«-W2<?+5cd+5te-6y. Sa*/3 =-d»c + 36(;2 + 62(f-5cd-Je + 5/. 2o»/3« ^-bc^ + ih^d + cd-bbe + bf. La*fiy = - 6«<? +. 2cd +be- bf. Za^^y = - (?rf + 3ie - 5/*; Za^fiyS = -be + bf; Za^it = -/. VI. 2o« = 5« - 66*<j + 96V _ 2c8 + 66«(f - I2bcd + 3<P - G^^e + 6ce + 66/"- %. Za*^ = 6*c - 462<^ + 2c» - 6»£? + 76cd; - 3# + d^e - 6ce - 6/*+ 6^. La*^ = 62c2 - 2c» - 263rf + 46crf - 3cP + 262e + 2ce - 66/+ 6y. Xo8/3» = c3 - 36c<; + 3rf2 + 362e - 3ce - Zbf+ Zg. Za*/3y = ft^c^ - Zbed + 3rf2 - 6^6 + 2ce + 6/- 6^. 2a»/3«y = 6crf - 3^2 - Si^c-f 4ce + 76/ - 12^. Xa2/32y4 = ^2 - 2ce + 26/- 2^ ; Xa»^y5 = 6^6 - 2<?« - 6/^ + S^r. Za^^yd = ce - 46/*+ 9^j Xa'^yle = 6/- 6^j Za^yieJ =flr. SYMMETKIC FUNCTIONS. 351 VIL Za^ = - ^r + 7^<. - 1468^2 + 7ftc» - 7b*d + 21h'»ed - 7c«d - 7bcP + 74»e - 14*ce + 7<fe - 7bY+ 7c/+ 7hg - 7h, - 7<fe + *»/- 7^/*- bg + 7&. 2:o»|3« = - iV + 36c» + 26*rf - 6^<?<; - 8c»rf + 7M* - 26»tf + 4*«! - 7(fe + 26y + 3c/- 7i^ + 7h, Zo*/8» = - 6c» + 3&«c<? + c»d- 66cP - 86»e + 2*c« + 6<fe + 76'/- 7cy - 7bg + 7h, £a»/3y = - b*d + 4b*cd ^ 2(^d - Ud* + b^e- Sbee + 7de- by + 2^+ ftgr - 7A. i:o*/3«y* =-Wjrf+2c»rf + W« + 8d»e-8*c< + 2efo-8fty + 4c/*+%-ia. Xo»/8»y = - c«<?+ 2W + ace - Srfe - 4dy+ 7c/+ % - 7A. 2:o«/32y« =-ft<P + 2acc + £fe-2a2/-3c/+7«sr_7A. 2o*^yd = - d»e + 36c« - 3(ie + 6y- 2cf- bg + 7h, i:a*^yd =-bce + Sde + ib'f- Gcf- %g + 21A. Za^P'y^ = - <fe + 3c/'- 6d^ + 7A ; la'^ya* = - 6*/ + 2c/*+ i^r - 7A. 2:a«/3»ya« = - 0/"+ 5«sr - 14& ; Za^^y^l = -bg + 7h; Lafiyitln = - *. VIII. Za» = 6» - 8i«c + 206*c2 - 16&V + 2c* + 8^d - 32a»crf + 2Abc^d + 12*2^2 _ 8ccP - 8a*c + 24*«c< - 8c«« - IQbde + 4e« + Sb»f - 166c/ + Sd/- 8b^g + Scg + Sbh - Si. ZaTfi zzb^c- 6i*c» + 96V - 2c* - 6*^: + ll6»crf - 176c»rf- 66»cP + 8c£P + 6*e- lOft^ce + Sc^c + 96cfe - 4e« - 6»/+ ^cf-- Bdf + h^g - %cg - 6A + 8i. Xa«/3* = 6*c« - 462c» + 2c* - 26«<? + mcd - 96«<P + 2ccP + 26*e - 66«ce - 4c2c + 166(fe - 4e« - 26y+ 46</- 84/^+ 2% + 4c^ - 86A + 8». So*^* = 6V - 2c* - 36»c<i + QbcH + 3*2cP - 7c£P + 36*e - Wee + ScH + W6 - 4e« - 86y+ 6^ + 74^ + 86V - 8c^ - 86* 4. 8». La*^* = c* - 46c«d + 26«rf« + 4crf« + 46«cc - 4c2e - Sbde + 6c« - 46»/ + 86c/- 4cjf + 462gr - 4c^ - 46A + 4». Sa'/S^y = 6*d - 66»crf + 56c2rf + bb^d^ - SceP - 6*6 + 46%6 - 2<^ - 96(fe + 46* + 6y- 36c/+ 84^- 6V + 2c5r + 6* - 8t. Xo»/3«y = 6»c£? - 36c2rf - 6«rf« + 6cflP - 36*e + 1162ce - 4c«e - 106ifo + 8«* + 36»/- 86c/+ df- 36V + ^g + 96* - 16». 2:o*/3»y = bi^d - 262<P - crf« - 6*cc + 106cfo - 8tf« + 46»/- 106c/*+ <J^- 96^ + 16c^ + 96* - 16». 5:a*^y« = 62rf«- 2c(P - 26«ce + 4c«c - 4e* + 26»/- 46c/ + SiJ/'- 26^ - 4c5r + 86A - 8«. Zo»/3»y2 = crf2 - 2c^ - 6(fe + 4e- + 56c/- 7dJ^ 66^ + 2c5r + 86* - 8». 352 TABLES. X^^ - Me - 4W» + 3cV + ^bde - 4e« - J*/ + 8ic/ - S^^-pft^ Xa*/3*yi =:6Ve-2<j«e-W« + 4^-4AV + llftc/-94^+4%-6(Jsr - 166ik + 24t. Za»/3Va =c*e-2W« + 2e*-ic/+8<{f+65V-9cy-56Jl + 12». X«»/3«y«a = W« - 4e« - 3ftc/+ 6<?/+ 6A«^ - 176* + 24». Sa«/3«y«a« = e« - 24/"+ 2cy - 26* + 2t. Lo^/Syat =6"/-36r/ + 84^-6V + 2«flr+6Jl-8t. 2:«»/3«yai = hqf- M/- 66^ + 8^^ + H** - 82t. 2a«/3«y*at = rf/*- 4<y + 96A - 16» ; Sa»/3ya«J = h^g - 2cg -bh + 8i. Xa«/3«y*«J= <SF - 66* + 20f ; So«/3yaeJn = 6* - ^t ; LajSyatJii© = i. IX. Sa» = -6» + 967c-276»<?« + 306V- 96c* -96«rf + 456«c<2-546^2rf + 9c«rf «■ 186»i» + 276«P - 8<P + 96*« - 366»c« + 276<?'^ + 276«cfe - IScde - 96e« - 96*/+ 276V- ^<^/- ^^W + 9«/+ 96»^ - 186<J^ + 9(%r - 96«* + 9c* + 96t - %*. Xa»/I = - 6'c + 76*c« - 146»c» + 76(?* + 6«rf - 136*cd + 3062(^d - 9c»rf + 66»rf» - 196ce^ + 8(? - 6*e + 126^ - 1960*6 - 116«<fe + 18c<fe + 56e« + 6*/- 116V+ 9^'/+ '^^W - 9e/- 6»^+ 106c^ - 9(%r + 6«* - 9c* -6t + 9/. 2a'/3« = - 6»c2 + 66»c» - 56c* + 26«rf - 106*crf + bb'^Ci^d + 5c»<? + IWd^ - ISbccP + 3rf» - 26»e + 86»ce + 6c«e - 206«<fe + 4c(fe + 96e« + 26*/- 662c/- 5c2/+ 186<^- 9e/ - 26»5r + 46c^ - 9^^ + 26'* + 6c* - 96* + 9;. Ea«/3« = - 6»c» + 36c* + 36*c<? - 96V<? - 3c«rf - 3b^cP + ISbcd^ - 6d» - 36*e + 126»ce - 96c2e - 96^^ + 96e2 + 36*/- 96V + 9cy- £le/- 36V + ^^ff + 9^^* - 9^* - 9** + 9/. £a*/3* = - 6c* + 46Vd + c^d - 2l^d^ - 7bcd^ + 3^ - 46»c« + 360*6 + ISb^de - 2cde - llhe^ + 46*/- 76V- cV- 26^^ + lie/- 96V + 186c!7 - 9^ + 96** - 9c* - 96i + ^. 2a^/3y =i-1fid+ Qb^ed - ^}^c?d + 2(^d- 66»^ + 126c^ - 3<i* ■\-¥€- Wet + 56c2c + 1162cfe - lUde - 566* - 6*/+ ^Hf^:^^f- lObdJ + 9e/+ 6V - 36c5' + 9<^ - 6** + 2c* + 6t - 9;*. Sa'^y = - 6*crf + 462c2(f - 2c3rf + b^d^ - 76c^ + 3d» + 365<! - 146»ce + 126c2e + ISb^de - 4c(fe - 146e« - 36*/+ 116V- 4c»/ - 106^/+ 18€/+ 36V - 86c^ - 36«* + 4c* + 106i - 18;. So*/3»y = - b^d + 2c^d + 263^* - 46crf* + 3^^ + 6»cc - 26c^ - bb*dc + 2cde + 66e2 - 46*/+ 166V- Sc*/- 56e^- 2e/ + 46V - 10605^ - 106«* + 18c* + 106t - 18;. SYMMETRIC FUNCTIONS. 353 + ^bdf^ 11^+ 5ftV - 1*% + 9dg^ bb^h + 9cA + Wt - 9;. - Sidf - ^+ 25V - -^^^^ + 9d^ - 2ft«A - 6cJl + 9W - ^\ + 2bdf- 2e/+ W^ - 46c^ - H*** + 4cA + 18« - 1%*. - 3&2A + 3cA + 35t - ^'. Sa«^ya = - fifie + 66»c« - W<^ - &*«<fo + 6c(fe + 6*e« + 5*/- 46«^+ 2<jy + 464^- 9«/- b*g + Ueg - 3<%r + ft** - 2cA - M + §^ Xa»/3«yi = - 6»<?e + 850^ + 6«rfe - 6c<fe - *«« + 4**/- 16*V+ 6^'/+ 1564^ - Te/"- 45»flr + 11% - 9<i^ + 45** - Bch - IIM + 27;. Zo*/3»yi =-*tf^ + 25»<fe + «fe-6W + 6«c/-66<(/'+13€/-6*»^ + 185<y - 9d^ + 11*«A - 20cA - 116» + 27;. Za*^V* =-*'& + 2c<fe + ft*« + 3*V- 6cy- 25^^+ 3«/- 66^ + 126<^ - 9<%r + 66«A + c* - 19W + 27;. 2a«/3»y«i = - ccfo + 83e« ^ 3<?y- 46<(/'- 7«/- 7&y + 18<^ + 126«* - 13cA - 19M + 27y. Xa»/32y«i2= - 6e« + 2bdf+ ef- Thcg - ^dg + ^h -\- bch - %% + y. 2:a»/3yii = - **/+ 4ft«c/- 20*/- 4*<J^+ 4«/+ b^g-^g-^ Bdg - 6«A + 2eh + W - ^•. 2o«i32ye« = - iV+ 2oy + 6<ir- 4^+ 6h»g - 14ft<!S' + 12<%r - 6M + 80A + 12bi - 3^'. Za^^yU = - <?«/ + 264^- 2^+ % - 3d^ - 66*A + llcA + 6« - 1%*. Za*^y^dg = - 54^+ 4e/+ 4% - 9(%r - 96*A + 6cA + 80M - 64;. Za'fi^y^S^t = ^^+8dg^ 5eh + 7bi - 9;. 2:a*^ya«J = - 5V + ^beg - 3(%r + *«A - 2<fJl - W + 9;. Za*^y6tl = -beg + Zdg + 66*A - 10<f A - 13W + 46;. 2a«/32y*a«J= -dg + bch- Ubi + 80; ; 2:a»/3ya«Jn = - 6«Jl + 2c* + W - ^. Sa^^yiejij = - c* + 7W - 27; ; 2a«/3ydt Jn© = - 6* + ^'. X. 2a" = *»• - 106»tf + 365«tf» - 60**c» + 255V - 2c» + 106'd - 605»cJ + 1005»c«<i - 405<?»rf+ 265*cP - eOb*cd* + l^cP + 10W» " 105«» + 605«ce - eOft^c^e + 10c<e - 405S(2e + 60bcd« - IOd*e + 166«e» - lOfli* + 105y- 405»c^+ 805cy + 80^4^- 20c4f - 205e/+ 6/« - I06V + ^^ff - 10<^ - 205<^ + 10^+ 106»* - 205c* + lOdh - 1052i + lOci + 105; - 10*. = i^c - 85«ca + 205*c» - 165*c« + 2c» - b^d + Ub^cd - 465»c»<i + 315c8rf - 75*d« + 335«crf» - 16(fld» - 7bd^ + 5«e - 145«c« + 335^6 - 10<;*e + IBb^de - i2bede + lOd^e ZZ 3^. TABLES. + 116^- Sf*-¥¥g^ 12i«c^ + lOc^ + llhdg - 10«jy - 5»A + UftcA - lOrf* + « - 10o» - %• + 10*. 2a»/9P = Mc« - 6J*c» + 9»c« - 2c» - 2*'xi + l26»od - 126»c»(l - 86cm •- 186«<9 + 285fc<9 + <^<^ - 106(^ + 26^; ~ 103«ce + 4&*(^ + €c«e + 24^(28 .- 28d6de + 10<^ - 11&%* + 2c^ - 26»/+ 8Wc/+ 2*<y- 22i«df + 4a{^+ 206e/ - 6/* + 2JV - 6W<^ - 6cV + 20*(%r - 10^ - 2J»A + 4beh - 10<;A + 2M + 6fi - 10^' + 10A-. Ea^/9» = JV - 4^V + 2tf» - 8*»crf + 12ih»d - 2ftc"rf + 36*d» - 24««cd« + 6c«d» + 11*^ + S*«« ^ 156*ce + im<*€ - 10cj»e + 126»<fc + ZbceU - lld»e - 166V + lOc^ -^ 86»/+ 12lfie/ - 96tf«/ - 96«4^- cdf+ 206c/- 6/« + 86^ - 96«c^ + lOc^ -* bdg - lOey - 86»A - 6cA + ll<tt + 106H - lOct - 106; + lOAr. l^/3« = 6V - 2c» - 46»Ai + 86c»rf + %b^cfi - 9d»d» + 26rf» + 46*<?« - 126*c«tf + 10<j«tf- 86»rfe + 126c<fo -r- 2d»e + 96«^ - 14cc« r- 46»/+ je6«c/- 186c«/r- 66»(i^ + 20edf ^ 46e/- 5/» + 46V ~ 6^ - 2(^^ - 46<^ + 14^ -^ 1063A 4- 206cA rr lOdh + 106*» - lOct - 106/ + lOife. £a»/3* = ^ - bb<fld + M^tfi + b<^d^ - 66<«» + 66«d«6 - 6«V - &6^d€ - 66<?(fo + 6<^ + 66V + $efi'^ mcf-\- 106c«/ + 106«<y - 16c4^^ 166e/+ lOy* + 66V - 1^6^^ + 6<jV + 106<fy - 6*^ - bm + 106cA - 8dA + 66«t - 6ct - 86; + bk. 2q«/3y = b^d - 76»tf<f + 146Vrf - 76c»<? + 76<rf« - 216«c<*« + 7c«d« + 76rf» - 6«« + 66*ce - 962<;*e + 2c»« ~ 186»rfe + 266<?<i« - 10d»e + 66«<j« - 6c«« + 6*/-^ 663^^ + 56(jy + 126«<^ - 12cdf- 116e/"+ 6/« - 6V + 46«c^ - 2cV - 116^ + 10«^ + 6»A - 36cA + lOdh -bH + 2ct + 6; - 10*. Xa^fify t= 6»crf - 66»c«rf + 56c»rf - b*d» + 96?od» - 7c«d» - 46^ - 86«c + 176*ce - 236?c«e + 4c3f - 166»ie + 216crfc + d«e + 176»«» ^ 12c^ + 36y- 14Wsf + 126cy + 136?(Jf - 8c^- 3166/" ^ lQ/1? - 36*5r + 116?<^ - 4<jV -^ lO**^' + 20^ + 36^* - 86<?A -r <^ -. 36«i + 4oi + 116; - 20*. ?:a«j3Py = b^'^d - 36c«rf - 26*(^ + 66V(^ + 3<j«<? - 76<i» - h*ce + 36V^ + b^de ^ 166«cfe + 13d»6 - 36 V + 4tf^4- 46»/- 196 V+ 186^/ + 166«(y-^ IGcfJ/*-- 766/"+ 10/^ - 46*5r + 186»^5r - 8cV - 46<^ - 4^ + 46«A - 106cA - rfA - 116«4 + 20ci + 116; - 20*. I5a»/3*y = 6<?»<? - Zb^ed^ - d^d^ + bbd» - 6V« + 66»(fe - 8<Pe - 86V + 4ce2 + 6»c/ - 6cy- 1262^+ 10cd/+ 236c/- 16/^ SYMMETRIC FUNCTIONS. 355 + 20dh - IIW + 2()c» + llbj - 20*. Z a«/9*y • = d*<P - 4fi»«? + 2<J«d« + 4W» - 2d*«e + 8 «?«6 - 4c^ - 8*«fo - 4*« - ft«e« + 10««« + 2Ay- 8*V'+ 4*<y + 104^ - 4c^ - 8bef + ^ - 2i>V + ^^ff - «^* - 2iy + 24"A - Ueh + lOdh - 2« - 6ct + lOi; - 10*. 2a»j3»y« = Wtfd» - 2(J«<P - W» - 2iV% + 4c«fe - «»<fe + SBcd6-^d*$ + iV - 12ce« + 6*^"- 186cy- 4*«^+ 17tf(J/'+ 106^~ Uf* - 5^^ + 153 V - 4<^^ - 193^ + 20i^ + 6^^ - 96ch - <tt - 12621 + 4c» + 20y- 20*. Zo*i3*y» = c»<P - 2&P - 2<j«e + 4*c<fe + 2d»« - 8i«e« + 2#e» + 2b(^/ + 26«4^- 12c4r+ 46^+ kf* - 6i«<y + lOtf*^ + iidg - 14<j^ + 66»* - laidSf + lOctt - 6d«t + 2c» + lOy - 10*. 2a*/5»y« =*<«»- Bhede - d»« + 36^« + 2ce» + Bbd^f- ^4f+ cdf- Sbef + ^ - Bi^off -4e^ff-^ IBbdff -- 2€ff -¥ S6»h + bch ~ llcUi - low + lOci + 106; - 10*. Za^Pyd zzlfie- eb*ee + 96^>e - 2<:*« + 66S(2e - l^bffde + 8<;*e - 66V H. &fe» - 6»/+ 66V- 56cy- 66«4f + 5c<Jf + 116</'-. 5/^ + 6 V ~ 46 V + 2#V + 46flr^ " lOesi - 6** + Bbch - Bdh + 6«»-2«-^'+10*. Za«/3»y* = 6Vje - 46»c«« + 2<j»e - 6»* + 76c<fe - 3tPe + 6 V - 6c^ - 46^ + 196V- 176cy- 196*<j^+ I6cd/+ 186^- 15/* + 46*^ - 156«cy + 6cV + 156<^ - e:i!^- 46»* + 116<?* - 9dh + 46«t-6<?t-12%*+80*. 2:o*/3*j^ = 6«tf*e - 2fl»e - 26»(fe + 46<J<fo - 3(Pe + 26««* + 2c^ - 6V+ ^<^/ + 86V- 4<?<J^- 126^+ lOf* + 56V - 19**^^ + 10<?V + 15** - % - 56»A + 136cA - 9rfA + 126«t - 22ct - 12^' + 80*. 2a*/3*y* = ^e - 36<J<fo + 8£f2«+ 86«^ - 3<j«« - 6cy+ 26V+ «^- 8^ + §/■« + 6V - <iV - 86<^ + 9^ - 66^ + 176caiwl5«Jk 4'662i-llct-6%'+15*. Zo*i32y«a = 6»rfe- 36c^ + 3d?e - 6«e« + 2<««-36V+ Wc»/+ 26*^ - 13c£j^- be/+iqp + 56V - 176«<y + 4<jV + 18^ - 18€^ - 56»* + 126c* - 9dh + 662*+ 2c» - 216;'+ 80*. Za*^H =zhcd€^Bd'e'-Zb^tf' + ici^'~Bbdy+4b'd/+6cd/''6/* + 76«<J^ - 8cV - 156£%' -I- 12<y - 1-26»* + 216tf*'+ B'dh + 266*1 - 28c» - 426; + 60*. 2a»/3V* = rf*e - 2ce* -cd/+ bbef- hp + 4o*^ - 76<%' + 2e^ - 46ri + lli^A + 76*1 - 10a - 76; + 10*. 356 TABLES. + 4&A - lOdh + Wi + Bei^l(ybf + lOA;. + 7« + ci- 15^+16*. £d*/3ydt = *•/- 64^+ W<^/+ W«<{^- 6<?4r- 6ftf/^+ 6/^ - *V + ^^ - 2<jV - *W^ + 4iy + i»* - SJtfA + 8<fA - 6«t + 2« + ^* - lOife. Za»/3«yi« = 6V- W*!/"- **4^+ 5<4f + btf- b/^ - 66V + l^***?^ - S^V - 196<^ + 16ey + 66»A - 146cA + IWA - 6d«» + 8c» + 13^' - iOife. - 166cA + 12dh - 18W + 24« + 18^* - 40*. - 286cA + 18d% - 96>» + 4c» + 38^' - 60*. . Xa»/3^y«a« =:«{f-86^+§/'-4<jV + 6W^ + 96c*-24i?*-21W + 2&» + 886;-60*. £as/3>yt^c = ft*/"- §^ - Bbdff + Seff + bhch -^dh- 7W - 8<?t + 31^' - 40*. Za«/3»y«^«l« =/» - 2i5^ + 2rf* - 2c» + 23; ~ 2*. Xa»/5yaf J = 6 V - ***<y + 2cV + *^ - 4«^ - ^* + 36c* - 8<» + W -2c»-%*+10*. Za*/3*yafJ =Wjy-2<jV-6<^ + 4iy-66»*+176c*-15<a + 3W - 10« - 146; + 50*. Xo»/3«yat J = flV - 26ii^ + 2e^ - 6c* + 3d* + 76«t - 13c» - 76; + 26*. Xa«/3«y«ai$ =hdg-Ug- bbeh + 12d* + 1462t - 12c» - 466; + 100*. 2o«^7«a*iJ = e^ - 4d4 + Oci - 166; + 26*. Za^^yUln = 6»* - 36c* + 3dA - h^i + 2ct + 6; - 10*. 2a»^ya«$»I = 6c* - 3rfA - 76«i + 12c» + 156; - 60*. Za^P'y^Uln = dh- 6ci + 206; - 50* ; Sa»i3ya«$t|0 = hH - 2c» - 6; + 10*. Za'^yi Ac. = ci - 86; + 35* ; So^/Jy Ac. = 6; - 10*. Prof. Caylej noticed a certain symmetry in the coefficients of the preceding formnls, which may be more easily exhibited by using Hirsch's notation. If such a snm as La*^y^iil be denoted [3*221'] and the coefficients be a„ a^f Ac, so that (32^1') will denote a^a^^ai*, then the formulsB for the sums of the fourth order may be written -1 -2 + 1 e« r-l C4 + 2 -2 + 1 -4 + 1 + 1 [4] =-4 [31] = + 4 [22] = + 2 [21«] = ^ 4 [1*]= + 1 The first Unc is to be read Sa* = -'ia^ + 4a,o, + 20^^ — 4a^|< + Ci*, and so on for the rest. Now what Prof. Gayley has proved is, that when the formnlsB already given are thus written, the figures are the same whether we read according to the rows or columns. The same thing holds for Prof. Cayley's (Phil. Trans., 1867, p. 489) formnlss expressing the coefficients (4), (31), Ac. in terms of the soms [4], [31], Ac. ( 357 ) INDEX. Absolute inTariants, 111, 175. Apolarity, 224, 336, 337, 346. Aronhold, on symboli^d methods, 147. On invariants of ternary cubic, 343. On the differential equations of in- rariants, 344. Baltzer, on determinants, 338. Bezout, on elimination, 81, 105, 338, 342. Bezoutiants, 343. Binet, on determinants, 338. Boole, on linear transformations, 109, 343. Form for the resultant of two quad- ratics, 24, 165, 180. Borchardt, proof that the equation of the secular inequalities has all its roots real, 54. Bordered Hessians reduced, 17. Bordered symmetrical determinants, ralue of, 86. Skew symmetrical determinants, 266. Brill, on sextic, 273. Brioschi expresses differential equation of . invariants in terms of roots, 342. On solution of the quintic, 257. On determinants, 339. Bumside, investigation of radius of sphere circumscribing tetrahedron, 26. Transformation of binary to ternary forms, 172. Applications of this method, 181, 189, 200, 202, 214, 222, 281. On solution of biquadratic, 196. Canonical forms, 150, 194, 228, 277, 281,345. Cauonizanta, 154, 22{), 282. Catalecticants, 156, 190, 261. Cauchy, on determinants, 33, 338, 340. Cayley (see also p. 343). His expression for relation connecting mutual distances of five points on a sphere, 26. of five points in space, 27. Application of skew determinants to the theory of orthogonal substitu- tions, 41, 339. Calculation of number of terms in a symmetrical determinant, 45. On symmetric functions of roots of equation, 58, 342. Statement of Bezout's method of elimination, 83. General expression for resultants as quotients of determinants, 87, 306. Notation for quantics, 99. Cayley, discovery of invariants, 109, 343. On the number of invariants of a binary quantic, 132, 176. Definition of oovariants, 136. Symbolical method of expressing in- variants and covariants, 137. Identifies two forms of canonizant of equations of odd degree, 155. On discriminant of discriminant, 166. On tact-invariants, 169. Method of forming a complete system of covariants, 176, 186, 199. Relation connecting oovariants of cubic, 186. Solution of a cubic, 186. Solution of a quartic, 195. On criteria of reality of roots, 197. On covariants of system formed by quartic and its Hessian, 201. On covariants of quintic, 237. Canonical form for quintic, 238. Tables of Sturmian functions, 240. On Tschimhausentransformation,251. On rational functional determinants, 340. On invariants of sextic, 345. Tables of symmetric tunctions, 350,. 356. Clebsch, on symbolical methods, 147. Special form of ternary quartics, 151, 335. Conditions a sextic may be sexti- covariant of quartic, 282. Proves that every invariant may be symbolically expressed, 315. Proves number of forms is finite, 320. Investigates resultant of quadratic and general equation, 326. Greneral expression for discriminant, 326. Investigates equation of system of inflexional tangents to a cubic, 330. On type form of even binary, 336, Form of resultant of two cubics, 345. Cockle, on the solution of the quintic, 257. Combinants, 161, 345. Invariant of invariant of « + Xv is a combinant, 211. Of a system of two quartics, 219. Common roots determined, 91. Commutants, 839. Complete system8,177,185,199,237,260,347. Concomitants, 121. Conditions that equations should have two common factozB, 78, 97, 291, 312. 358 INDEX. ConditkmB for syiteiDB of equalities be- tween roots, 126. That qaandc should be reducible to sum of powers, 156. That « + Xv should have a cubic factor, 166, 205, 221, 279. That « + Xo hare two double factors, 166, 220, 279. That four points should form a har- monic system, 179. That three pairs of ^ints shonld form system in inTolntion. 180. For three qnadrics to oe differentials of a qnartic, 224. to be Qoadric ooyariants of two cnbics, 225. That qnartic shonld hare two square factors, 220. That two quartics should be differen- tials of same quintic, 220, 228. That quintic shonld admit of being brought by linear transformation to Jerrard*s form, 282. That quintic should have two square or a cubic factor. 288. That sextic shouki hare two square or a cubic factor, 268. That roots of sextic should be in in- Tolution, 270. For sextic to be Hessian of a qninti(^ 282. For sextic to be sextiooTariant of qnartic, 282. That qnantic have two square or one cubic factor, 294. Conjugate forms, 226, 887, 846. Contragredience, 118, 846. Contravariants, 117, 120. Of binary quantics not essentially dis- tinct from oovariants, 127. Continuants, 18. Covariantfi. 114; distinct. 175. How defined bv Cayley, 136. Number of, for binary quantic,182,176. Cramer, on determinants, 888. Critical functions, 60. CubicoTariant of cubic, 180, 183. Cubic discussed, 188. Bystom of twa 204 ; of four, 215,837. Cubic, quaternary, its canonical form, 160. Cubiuvariants, only type of, 141. Doriratiyos of derivatiyes expressed sym- bolically. 821. Dialytio method of elimination, 79, 842. Differential coefficients of determinants, 85. Of roAultants with respect to quantities entering into all the quantics, 96. Difforcntial conation of functions of differ- euoos of roots, 61, 842. Of invariants, 131. IMfTorcntiution mutual, of covariants and rt^ntravnriantji, 126. Discriminant defined, 99. Of binary nnantic expressed as deter- minant) 33, Discriminant of product of two qualities^ 101, 167. Of discriminants, 166. Sign of distinguishes whether equation has even or odd numbers of pairs of imaginary roots, 239. General symbolical expression for, 826. Distinct invariants and covariants, 175. Double' points of involution, 162. Double tangents of plane curves, 338. D'Ovidio, on two cubics, 213 ; quartics, 345. Durf ee, on symmetric functions, 350. Eisenstein, expression for general solntioa of qnartic, 343. Eliminants defined, 66. Elimination, 67. Emanants, 115, 333. Equalities between roots of an eqnatloB, conditions for, 125. Equianharmonic, 184. Invariant, 190, 259. Eukr, a theorem of, 25. On the theory of orthogonal substi- tutions, 44. On elimination, 77, 342. Evectants, 122. Of discriminant which vanishes, 1!^ Symbolical expression for, 146. Of discriminant of cubic, 130, 183. Of cubinvariant of qnartic, 192. Of quartinvariant of quintic, 237. Of sexUc, 279. FaH de Bruno, calculates invariant el quintic, 230. On elimination, 342. On symmetric functions, 350. Forme-type of quintic, 249. Of even (^uantic, 335. Of quantic of oider 3p, 337. Gauss, on linear transformations, 338, 343. Gerbaldi, on two cubics, 213. Gordan, on number of covariants, 17^ 213, 224, 260, 320, 324. Gnndelfinger, on system of cubic and qnartic, 218. Harley, on solution of a quintic, 257. Harmonic invariant, 190, 260. Hermite, law of reciprocity, 142, 179. On transformation of a quadratic function, 42. On concomitants of system formed by quartic and its Hessian, 201. Canonical form for quintic, 233. Discovery of skew invariant of quintic 233, 345. Expression by invariants of oonditiona of reality of roots, 242, 250. Forme-type of quintic, 249. On TschuBhausen transformation, 251. Solution of quintics by elliptic func- tions, 257. Expression of invariants of quintic in terms of roots, 258. INDEX. 359 Hefldans, 17, 1 17, 144, 183, 192, 227, 261,841 . Cbntain all square facton of original qaantic, 153, 198. Of Hessiaiis, 202, 821. Hill, on quintic, 260. Hirsch's tables of symmetric functions) 850. Homographic determinant, 33. Byper-determinantB, 848. Inflexional tangents to cubic, calculation of their equation, 830. Inyariants, 111 ; irreducible, 176. Absolute, 111, 190. Skew, 131. How many independent^ 111. Belation connecting weight and order of, 180. InTolution, 162. Condition roots of sextic be in, 270. Determinant, 25, 88, 180, 270. Foci of, 179, 193, 259. Jacobi, on determinants and linear timns- formationa, 839, 340, 341. Jacobian, of systems of equations, 84, 117, 842. Properties of, 84. Geometrically interpreted, 162. Its discriminant discussed, 164. Of two q^uadratics, 179. Of quartic and its HessiaiL 201. Of systems of quartics, 271, 274. Jemu^l, transformation of a quintic, 282, 250, 259. Joachimsthal, ezpreasion for area of a triangle inscribed in an ellipse, 26. Theorem on form of discriniinant, 102. Eronecker, solution of quintic bj elliptic functions, 257. Kummer's resolution into sum of squares of discriminant of cubic which de« termines axes of a quadric, 55. Lagrange, on solution of quintic, 267. On dstenninants. 888. On conditions that equation should - have two pairs of ec^xul roots, 842. On linear transformations, 348. Laplace, on determinants, 838. E<quatk>n of secular inequalities, 48. Leibnitz, his claim to inyention of deter- minants, 888. lindemann, on geometric exposition, 174, 216, 218, 886. Condition for sextic to be Hessian of a quintic, 283. Linear ooYariants of cubic and quadratic, 187. Of two cubics, 209, 215. Of cubic and quartic, 219. Of quintic, 285, 249. ^acMahon, on s^pimetric functions, 850; Meyer, on apolarity, 224, 276. Minor determinai^ts, 10, 28. Minor determinants, of reciprocal system ho^ related to those of original, 81. Muir, on continuants, 18. Multiplication of determinants, 20. Newton, on sums of powers of roots of equation, 56. Number of terms in a symmetrical de- terminant, 45. Of quadrics which can be described through five points to touch four planes, 298. Of invariants of a binary quantio, 132, 175. Of distinct forms finite, 176, 825.. Order of determinants, 7. Of symmetric functions, 58. Of invariants, 180. Of resultant of any equations, 76. Of discriminants, 99. Of systems of equations, 284. Orthogonal substitutions, 42. Osculants, 171. Poisson's method of forming symmetric functions of common roots of sys- tems of equations, 72, 842. Quadratic forms, transformed, 42. Reducible to sum of squares, 151. Number of positive and negative squares fixed, 151. General expression for resultant with equation of n<* degree, 827. Quadric systems, 179. Quadrinvariants of binary quantics, 128. Quartic, theory of, 189. S^^;em of two, 219, 271. Quartinvariant of odd quantic, 128. Quintic, theory of. 227, 847. Involved in tneory of sextic, 278. Reciprocal determinants, 80. Reciprocity, Hermite's law of, 142, 179. Reducing sextic for quintic, forms of, 257. Resultant, order and weight of, 66, 75. Of two quadratics, 68, 77, 88, 85, 180. Of two cubics, 77, 85, 206, 346. Of two quartics, 81, 86, 220, 277. Of quadratic and any equation, 827. Tables of, 348. Reve, on apolarity, 846. Rob^i», MichaeL on sources of oovaiiants. 184, 238. ^ On application of Sturm's theorem to qumtics, 240. On equation of squares of diffeienoee, 847. Roberts, bamuel, on orders of systems of equationsJdlO. Roberto, W. R. W., on twisted cubic, 174, 219c, d. ' Rodrigues, orthogonal transformations, 44. Rosanes,on conjugate forms, 346. 300 INDEX. ^Jl \?1^ SdilMiiigSy on ooojiigaitB fonnsy 225. SemmTaiuuit^ 176. *■ notatkm for differential equatioii of oormrianta, 65. theory <^ S60, S47. symmetric determinanta, of eren d^pnee are perfect aquarea, 38. Bofdeiedy 266. Skew inTaiisnta defined, 131. 'Of all qaantica Tanish whoi qnantic wants alternate terms, 234. Skew inyariant of qnintic, 233, 282. Yanisbes if qnintic can be Hnearlr transformed to recarriDg form, 242. Expression in terms of roots, 258. Skew inrariant of sextic, 260, 269, 282. Source of ooyariants, 134, 238. Sphere dicnmacaribing tetrahedron, 26. Belations connecting mntoal distanoea of pcunta on, 26. Spottiswoode, on determinants, 338. Stephanos, on sextic, 273, 282. Stroh, on two qoartics, 345. Storm's functions, Sylvester's expreaaionB for, 49. In case of qoartic, 197. of qointic, 240. Extension of, 105. Storm, B., on geometrical representation, 845. Snperflaoos yariable, method of osing, 207. SylTester (see also p. 343). Umbral notation for determinants, 8. Proof that equation of secolar in- equalities has all real roots, 28, 48. Expression for Sturm's functions in terms of roots, 49. Dialytic method of elimination, 79, 342. Expression of resultant as determi- nant, 86. Sylteater,exten8ion of Storm's theorem,106« On nomenclature, 121, 122, 190, 841, 342, 345. Canonical forms of odd and eren degrees, 153, 156, 227, 346. Of quaternary culnc, 160. Expressions for discriminant with re- gard to variables which do not enter expKcitly, 168. On oscnlante, 171. On number of distinct forms, 176. Beduoes system of two cubics, 213. Beduces system of cubic and quartic, 218. Beduoes system of two quartica, 224, 345. Investigation of invariant conditionB for r«dity of roots of quintic, 242. On Bezoutiants, 343. On combinants, 345. Symbolical expression for InyariantB, dkc. 137,314. Symmetric functions, 56. Their use in finding inTariants^ 124. Tables of, 350. ■ Tact-invariants, 169. Of complex curves, 170. Tetrahedron, radius of circamflcribing sphere, 26. Transvection and transvectants, 320, 346. Tschimhausen, transformation of equa- tions, 250. Type, or typical forms, 249, 335. Umbral notation, 8, 314, 338. Yandermonde, on determinapta, 338. Warren, on system of two qoartics, 221. THE END. \ \^ { \:^r W. MBTCALFB AND BON, PRINTERS, CAMBBID6B. X \ \